Non-associative algebra

<h2 id="algebras-satisfying-identities">Algebras satisfying identities</h2>
Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study.
For this reason, the best-known kinds of non-associative algebras satisfy <a href="/facts/Identity_(mathematics)/LR46j4uR">identities</a>, or properties, which simplify multiplication somewhat.
These include the following ones.

<h3>Usual properties</h3>
Let x, y and z denote arbitrary elements of the algebra A over the field K.
Let powers to positive (non-zero) integer be recursively defined by x1 ≝ x and either xn+1 ≝ xnx<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> (right powers) or xn+1 ≝ xxn<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> (left powers) depending on authors.

<ul><li><a href="/facts/Unital_algebra/8T8ulWJb">Unital</a>: there exist an element e so that ex = x = xe; in that case we can define x0 ≝ e.</li>
<li><a href="/facts/Associativity/EQX8lV6r">Associative</a>: (xy)z = x(yz).</li>
<li><a href="/facts/Commutativity/WaYxJLxd">Commutative</a>: xy = yx.</li>
<li><a href="/facts/Anticommutative/N2xrAKH5">Anticommutative</a>:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> xy = −yx.</li>
<li><a href="/facts/Jacobi_identity/C9HoyQ3z">Jacobi identity</a>:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> (xy)z + (yz)x + (zx)y = 0 or x(yz) + y(zx) + z(xy) = 0 depending on authors.</li>
<li><a href="/facts/Jordan_identity/4BBGsQd9">Jordan identity</a>:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> (x2y)x = x2(yx) or (xy)x2 = x(yx2) depending on authors.</li>
<li><a href="/facts/Alternative_algebra/3boGwrqQ">Alternative</a>:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> (xx)y = x(xy) (left alternative) and (yx)x = y(xx) (right alternative).</li>
<li><a href="/facts/Flexible_algebra/JPpgC0kU">Flexible</a>:<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> (xy)x = x(yx).</li>
<li>nth power associative with n ≥ 2: xn−kxk = xn for all integers k so that 0 < k < n.
<ul><li>Third power associative: x2x = xx2.</li>
<li>Fourth power associative: x3x = x2x2 = xx3 (compare with fourth power commutative below).</li></ul></li>
<li><a href="/facts/Power_associative/192PkK5G">Power associative</a>:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> the subalgebra generated by any element is associative, i.e., nth power associative for all n ≥ 2.</li>
<li>nth power commutative with n ≥ 2: xn−kxk = xkxn−k for all integers k so that 0 < k < n.
<ul><li>Third power commutative: x2x = xx2.</li>
<li>Fourth power commutative: x3x = xx3 (compare with fourth power associative above).</li></ul></li>
<li>Power commutative: the subalgebra generated by any element is commutative, i.e., nth power commutative for all n ≥ 2.</li>
<li><a href="/facts/Nilpotent_algebra/gRC3q2tz">Nilpotent</a> of index n ≥ 2: the product of any n elements, in any association, vanishes, but not for some n−1 elements: x1x2…xn = 0 and there exist n−1 elements so that y1y2…yn−1 ≠ 0 for a specific association.</li>
<li><a href="/facts/Nil_algebra/gRC3q2tz">Nil</a> of index n ≥ 2: power associative and xn = 0 and there exist an element y so that yn−1 ≠ 0.</li></ul>
<h3>Relations between properties</h3>
For K of any <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a>:

<ul><li>Associative implies alternative.</li>
<li>Any two out of the three properties left alternative, right alternative, and flexible, imply the third one.
<ul><li>Thus, alternative implies flexible.</li></ul></li>
<li>Alternative implies Jordan identity.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></li>
<li>Commutative implies flexible.</li>
<li>Anticommutative implies flexible.</li>
<li>Alternative implies power associative.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>Flexible implies third power associative.</li>
<li>Second power associative and second power commutative are always true.</li>
<li>Third power associative and third power commutative are equivalent.</li>
<li>nth power associative implies nth power commutative.</li>
<li>Nil of index 2 implies anticommutative.</li>
<li>Nil of index 2 implies Jordan identity.</li>
<li>Nilpotent of index 3 implies Jacobi identity.</li>
<li>Nilpotent of index n implies nil of index N with 2 ≤ N ≤ n.</li>
<li>Unital and nil of index n are incompatible.</li></ul>
If K ≠ <a href="/facts/GF(2)/KIKKpYua">GF(2)</a> or dim(A) ≤ 3:

<ul><li>Jordan identity and commutative together imply power associative.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a><a class="footnote-ref" id="fnref:26" href="#fn:26">26</a></li></ul>
If char(K) ≠ 2:

<ul><li>Right alternative implies power associative.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a><a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>
<ul><li>Similarly, left alternative implies power associative.</li></ul></li>
<li>Unital and Jordan identity together imply flexible.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a></li>
<li>Jordan identity and flexible together imply power associative.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a></li>
<li>Commutative and anticommutative together imply nilpotent of index 2.</li>
<li>Anticommutative implies nil of index 2.</li>
<li>Unital and anticommutative are incompatible.</li></ul>
If char(K) ≠ 3:

<ul><li>Unital and Jacobi identity are incompatible.</li></ul>
If char(K) ∉ {2,3,5}:

<ul><li>Commutative and x4 = x2x2 (one of the two identities defining fourth power associative) together imply power associative.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a></li></ul>
If char(K) = 0:

<ul><li>Third power associative and x4 = x2x2 (one of the two identities defining fourth power associative) together imply power associative.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a></li></ul>
If char(K) = 2:

<ul><li>Commutative and anticommutative are equivalent.</li></ul>
<h3>Associator</h3>
Main article: <a href="/facts/Associator/qglwnfem">Associator</a>
The associator on A is the K-<a href="/facts/Multilinear_map/YosSfBKE">multilinear map</a> 
 
 
 
 [
 ⋅
 ,
 ⋅
 ,
 ⋅
 ]
 :
 A
 ×
 A
 ×
 A
 →
 A
 
 
 {\displaystyle [\cdot ,\cdot ,\cdot ]:A\times A\times A\to A}
 
 given by

[x,y,z] = (xy)z − x(yz).
It measures the degree of nonassociativity of 
 
 
 
 A
 
 
 {\displaystyle A}
 
, and can be used to conveniently express some possible identities satisfied by A.
Let x, y and z denote arbitrary elements of the algebra.

<ul><li>Associative: [x,y,z] = 0.</li>
<li>Alternative: [x,x,y] = 0 (left alternative) and [y,x,x] = 0 (right alternative).
<ul><li>It implies that permuting any two terms changes the sign: [x,y,z] = −[x,z,y] = −[z,y,x] = −[y,x,z]; the converse holds only if char(K) ≠ 2.</li></ul></li>
<li>Flexible: [x,y,x] = 0.
<ul><li>It implies that permuting the extremal terms changes the sign: [x,y,z] = −[z,y,x]; the converse holds only if char(K) ≠ 2.</li></ul></li>
<li>Jordan identity:<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> [x2,y,x] = 0 or [x,y,x2] = 0 depending on authors.</li>
<li>Third power associative: [x,x,x] = 0.</li></ul>
The nucleus is the set of elements that associate with all others:<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> that is, the n in A such that

[n,A,A] = [A,n,A] = [A,A,n] = {0}.
The nucleus is an associative subring of A.

<h3>Center</h3>
The center of A is the set of elements that commute and associate with everything in A, that is the intersection of

C
        (
        A
        )
        =
        {
        n
        ∈
        A
         
        
          |
        
         
        n
        r
        =
        r
        n
        
        ∀
        r
        ∈
        A
        
        }
      
    
    {\displaystyle C(A)=\{n\in A\ |\ nr=rn\,\forall r\in A\,\}}

with the nucleus. It turns out that for elements of C(A) it is enough that two of the sets 
 
 
 
 (
 [
 n
 ,
 A
 ,
 A
 ]
 ,
 [
 A
 ,
 n
 ,
 A
 ]
 ,
 [
 A
 ,
 A
 ,
 n
 ]
 )
 
 
 {\displaystyle ([n,A,A],[A,n,A],[A,A,n])}
 
 are 
 
 
 
 {
 0
 }
 
 
 {\displaystyle \{0\}}
 
 for the third to also be the zero set.

<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> R3 with multiplication given by the <a href="/facts/Vector_cross_product/uRBJzkcW">vector cross product</a> is an example of an algebra which is anticommutative and not associative. The cross product also satisfies the Jacobi identity.</li>
<li><a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a> are algebras satisfying anticommutativity and the Jacobi identity.</li>
<li>Algebras of <a href="/facts/Vector_field/ccFNuPPp">vector fields</a> on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> (if K is R or the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C) or an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> (for general K);</li>
<li><a href="/facts/Jordan_algebra/4BBGsQd9">Jordan algebras</a> are algebras which satisfy the commutative law and the Jordan identity.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a></li>
<li>Every associative algebra gives rise to a Lie algebra by using the <a href="/facts/Commutator/eYxzFmFY">commutator</a> as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.</li>
<li>Every associative algebra over a field of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> other than 2 gives rise to a Jordan algebra by defining a new multiplication x*y = (xy+yx)/2. In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called special.</li>
<li><a href="/facts/Alternative_algebra/3boGwrqQ">Alternative algebras</a> are algebras satisfying the alternative property. The most important examples of alternative algebras are the <a href="/facts/Octonions/nC2sIz3p">octonions</a> (an algebra over the reals), and generalizations of the octonions over other fields. All associative algebras are alternative. Up to isomorphism, the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions.</li>
<li><a href="/facts/Power-associative_algebra/192PkK5G">Power-associative algebras</a>, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras over a field other than <a href="/facts/GF(2)/KIKKpYua">GF(2)</a> (see previous section), and the <a href="/facts/Sedenion/DT3scdVK">sedenions</a>.</li>
<li>The <a href="/facts/Hyperbolic_quaternion/QwsGfTJ7">hyperbolic quaternion</a> algebra over R, which was an experimental algebra before the adoption of <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a> for <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>.</li></ul>
More classes of algebras:

<ul><li><a href="/facts/Graded_algebra/AIUjkSaP">Graded algebras</a>. These include most of the algebras of interest to <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>, such as the <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a>, <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a>, and <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> over a given <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. Graded algebras can be generalized to <a href="/facts/Filtered_algebra/jL7AG71G">filtered algebras</a>.</li>
<li><a href="/facts/Division_algebra/xp5Bepx3">Division algebras</a>, in which multiplicative inverses exist. The finite-dimensional alternative division algebras over the field of real numbers have been classified. They are the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> (dimension 1), the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> (dimension 2), the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> (dimension 4), and the <a href="/facts/Octonion/nC2sIz3p">octonions</a> (dimension 8). The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions.</li>
<li><a href="/facts/Quadratic_algebra/ADnCK0nY">Quadratic algebras</a>, which require that xx = re + sx, for some elements r and s in the ground field, and e a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions.</li>
<li>The <a href="/facts/Cayley%E2%80%93Dickson_algebra/aDmlkB9m">Cayley–Dickson algebras</a> (where K is R), which begin with:
<ul><li>the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C (a commutative and associative algebra);</li>
<li>the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> H (an associative algebra);</li>
<li>the <a href="/facts/Octonion/nC2sIz3p">octonions</a> O (an <a href="/facts/Alternative_algebra/3boGwrqQ">alternative algebra</a>);</li>
<li>the <a href="/facts/Sedenion/DT3scdVK">sedenions</a> S;</li>
<li>the <a href="/facts/Trigintaduonion/DT3scdVK">trigintaduonions</a> T and the infinite sequence of Cayley-Dickson algebras (<a href="/facts/Power-associative_algebra/192PkK5G">power-associative algebras</a>).</li></ul></li>
<li><a href="/facts/Hypercomplex_algebra/BwKwTBF2">Hypercomplex algebras</a> are all finite-dimensional unital R-algebras, they thus include Cayley-Dickson algebras and many more.</li>
<li>The <a href="/facts/Poisson_algebra/dRkCcexQ">Poisson algebras</a> are considered in <a href="/facts/Geometric_quantization/NGcSz0CM">geometric quantization</a>. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.</li>
<li><a href="/facts/Genetic_algebra/Eo2sqnAF">Genetic algebras</a> are non-associative algebras used in mathematical genetics.</li>
<li><a href="/facts/Triple_system/UkJ0udTN">Triple systems</a></li></ul>
See also: <a href="/facts/List_of_algebras/Aw5zWK60">list of algebras</a>
<h2 id="properties">Properties</h2>
There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a <a href="/facts/Zero_divisor/5GraAU8o">zero divisor</a>. For example, all non-zero elements of the <a href="/facts/Sedenion/DT3scdVK">sedenions</a> have a two-sided inverse, but some of them are also zero divisors.

<h2 id="free-non-associative-algebra">Free non-associative algebra</h2>
The free non-associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>
Kurosh <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> that every subalgebra of a free non-associative algebra is free.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a>

<h2 id="associated-algebras">Associated algebras</h2>
An algebra A over a field K is in particular a K-vector space and so one can consider the associative algebra EndK(A) of K-linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras of EndK(A), the derivation algebra and the (associative) enveloping algebra.

<h3>Derivation algebra</h3>
Main article: <a href="/facts/Derivation_algebra/CXbpIMgk">Derivation algebra</a>
A <a href="/facts/Derivation_(abstract_algebra)/vcSqpxzu">derivation</a> on A is a map D with the property

D
        (
        x
        ⋅
        y
        )
        =
        D
        (
        x
        )
        ⋅
        y
        +
        x
        ⋅
        D
        (
        y
        )
         
        .
      
    
    {\displaystyle D(x\cdot y)=D(x)\cdot y+x\cdot D(y)\ .}

The derivations on A form a subspace DerK(A) in EndK(A). The <a href="/facts/Commutator/eYxzFmFY">commutator</a> of two derivations is again a derivation, so that the <a href="/facts/Lie_bracket/n2gAUkNq">Lie bracket</a> gives DerK(A) a structure of <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a>

<h3>Enveloping algebra</h3>
There are linear maps L and R attached to each element a of an algebra A:<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>

L
        (
        a
        )
        :
        x
        ↦
        a
        x
        ;
         
         
        R
        (
        a
        )
        :
        x
        ↦
        x
        a
         
        .
      
    
    {\displaystyle L(a):x\mapsto ax;\ \ R(a):x\mapsto xa\ .}

Here each element 
 
 
 
 L
 (
 a
 )
 ,
 R
 (
 a
 )
 
 
 {\displaystyle L(a),R(a)}
 
 is regarded as an element of EndK(A). The associative enveloping algebra or multiplication algebra of A is the sub-associative algebra of EndK(A) generated by the left and right linear maps 
 
 
 
 L
 (
 a
 )
 ,
 R
 (
 a
 )
 
 
 {\displaystyle L(a),R(a)}
 
.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a> The centroid of A is the centraliser of the enveloping algebra in the endomorphism algebra EndK(A). An algebra is central if its centroid consists of the K-scalar multiples of the identity.<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>
Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>

<ul><li>Commutative: each L(a) is equal to the corresponding R(a);</li>
<li>Associative: any L commutes with any R;</li>
<li>Flexible: every L(a) commutes with the corresponding R(a);</li>
<li>Jordan: every L(a) commutes with R(a2);</li>
<li>Alternative: every L(a)2 = L(a2) and similarly for the right.</li></ul>
The quadratic representation Q is defined by<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a>

Q
 (
 a
 )
 :
 x
 ↦
 2
 a
 ⋅
 (
 a
 ⋅
 x
 )
 −
 (
 a
 ⋅
 a
 )
 ⋅
 x
  
 
 
 {\displaystyle Q(a):x\mapsto 2a\cdot (a\cdot x)-(a\cdot a)\cdot x\ }
 
,
or equivalently,

Q
        (
        a
        )
        =
        2
        
          L
          
            2
          
        
        (
        a
        )
        −
        L
        (
        
          a
          
            2
          
        
        )
         
        .
      
    
    {\displaystyle Q(a)=2L^{2}(a)-L(a^{2})\ .}

The article on <a href="/facts/Universal_enveloping_algebra/VJ3Syz4W">universal enveloping algebras</a> describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them. For Lie algebras, such enveloping algebras have a universal property, which does not hold, in general, for non-associative algebras. The best-known example is, perhaps the <a href="/facts/Albert_algebra/YThFsbVl">Albert algebra</a>, an exceptional <a href="/facts/Jordan_algebra/4BBGsQd9">Jordan algebra</a> that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_algebras/Aw5zWK60">List of algebras</a></li>
<li><a href="/facts/Commutative_non-associative_magmas/aAtjsaj3">Commutative non-associative magmas</a>, which give rise to non-associative algebras</li></ul>
<h2 id="citations">Citations</h2>

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Abraham_Adrian_Albert/whzaVMMg">Albert, A. Adrian</a> (2003) [1939]. <a href="https://books.google.com/books?isbn=0821810243">Structure of algebras</a>. American Mathematical Society Colloquium Publ. Vol. 24 (Corrected reprint of the revised 1961 ed.). New York: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1024-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0023.19901">0023.19901</a>.</li>
<li><a href="/facts/Abraham_Adrian_Albert/whzaVMMg">Albert, A. Adrian</a> (1948a). <a href="https://doi.org/10.2307%2F1990399">"Power-associative rings"</a>. <a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a>. 64: 552–593. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990399">10.2307/1990399</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990399">1990399</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0027750">0027750</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0033.15402">0033.15402</a>.</li>
<li><a href="/facts/Abraham_Adrian_Albert/whzaVMMg">Albert, A. Adrian</a> (1948b). "On right alternative algebras". <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. 50: 318–328. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969457">10.2307/1969457</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969457">1969457</a>.</li>
<li>Bremner, Murray; Murakami, Lúcia; Shestakov, Ivan (2013) [2006]. <a href="https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf">"Chapter 86: Nonassociative Algebras"</a> (PDF). In Hogben, Leslie (ed.). <a href="https://books.google.com/books?isbn=9781498785600">Handbook of Linear Algebra</a> (2nd ed.). <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-498-78560-0.</li>
<li><a href="/facts/Israel_Nathan_Herstein/fWts4GeY">Herstein, I. N.</a>, ed. (2011) [1965]. <a href="https://books.google.com/books?isbn=3642110363">Some Aspects of Ring Theory: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna (Como), Italy, August 23-31, 1965</a>. C.I.M.E. Summer Schools. Vol. 37 (reprint ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-6421-1036-3.</li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (1968). <a href="https://books.google.com/books?isbn=9780821846407">Structure and representations of Jordan algebras</a>. American Mathematical Society Colloquium Publications, Vol. XXXIX. Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-821-84640-7. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0251099">0251099</a>.</li>
<li>Knus, Max-Albert; <a href="/facts/Alexander_Merkurjev/DowcduN3">Merkurjev, Alexander</a>; <a href="/facts/Markus_Rost/Qcwhdrp7">Rost, Markus</a>; <a href="/facts/Jean-Pierre_Tignol/FPIXDwWj">Tignol, Jean-Pierre</a> (1998). <a href="https://books.google.com/books?isbn=0821809040">The book of involutions</a>. Colloquium Publications. Vol. 44. With a preface by J. Tits. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0904-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0955.16001">0955.16001</a>.</li>
<li>Koecher, Max (1999). Krieg, Aloys; Walcher, Sebastian (eds.). <a href="https://books.google.com/books?isbn=3540663606">The Minnesota notes on Jordan algebras and their applications</a>. Lecture Notes in Mathematics. Vol. 1710. Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-66360-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1072.17513">1072.17513</a>.</li>
<li>Kokoris, Louis A. (1955). <a href="https://doi.org/10.2307%2F2032920">"Power-associative rings of characteristic two"</a>. <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>. 6 (5). <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>: 705–710. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2032920">10.2307/2032920</a>.</li>
<li>Kurosh, A.G. (1947). "Non-associative algebras and free products of algebras". Mat. Sbornik. 20 (62). <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0020986">0020986</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0041.16803">0041.16803</a>.</li>
<li><a href="/facts/Kevin_McCrimmon/kqkDt5YE">McCrimmon, Kevin</a> (2004). <a href="https://books.google.com/books?isbn=9780387954479">A taste of Jordan algebras</a>. Universitext. Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fb97489">10.1007/b97489</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95447-9. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2014924">2014924</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1044.17001">1044.17001</a>. <a href="http://www.math.virginia.edu/Faculty/McCrimmon/">Errata</a>.</li>
<li>Mikheev, I.M. (1976). "Right nilpotency in right alternative rings". <a href="/facts/Siberian_Mathematical_Journal/VxHRpvoL">Siberian Mathematical Journal</a>. 17 (1): 178–180. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00969304">10.1007/BF00969304</a>.</li>
<li>Okubo, Susumu (2005) [1995]. <a href="https://books.google.com/books?isbn=0521017920">Introduction to Octonion and Other Non-Associative Algebras in Physics</a>. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511524479">10.1017/CBO9780511524479</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-01792-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0841.17001">0841.17001</a>.</li>
<li>Rosenfeld, Boris (1997). <a href="https://books.google.com/books?isbn=0792343905">Geometry of Lie groups</a>. Mathematics and its Applications. Vol. 393. Dordrecht: Kluwer Academic Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-4390-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0867.53002">0867.53002</a>.</li>
<li>Rowen, Louis Halle (2008). <a href="https://books.google.com/books?isbn=0821884085">Graduate Algebra: Noncommutative View</a>. Graduate studies in mathematics. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-8408-5.</li>
<li><a href="/facts/Richard_D._Schafer/Y3qDuQ88">Schafer, Richard D.</a> (1995) [1966]. <a href="https://books.google.com/books?isbn=0486688135">An Introduction to Nonassociative Algebras</a>. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-68813-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0145.25601">0145.25601</a>.</li>
<li>Zhevlakov, Konstantin A.; Slin'ko, Arkadii M.; Shestakov, Ivan P.; <a href="/facts/Anatoly_Shirshov/76KreJCj">Shirshov, Anatoly I.</a> (1982) [1978]. <a href="https://www.researchgate.net/publication/260600596_Rings_that_are_nearly_associative">Rings that are nearly associative</a>. Translated by Smith, Harry F. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-779850-1.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schafer 1995, Chapter 1. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schafer 1995, p. 1. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Albert 1948a, p. 553. - Albert, A. Adrian (1948a). "Power-associative rings". Transactions of the American Mathematical Society. 64: 552–593. doi:10.2307/1990399. ISSN 0002-9947. JSTOR 1990399. MR 0027750. Zbl 0033.15402. <a href="https://doi.org/10.2307%2F1990399" target="_blank">https://doi.org/10.2307%2F1990399</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schafer 1995, p. 30. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schafer 1995, p. 128. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Schafer 1995, p. 3. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Schafer 1995, p. 3. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Okubo 2005, p. 12. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Schafer 1995, p. 91. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Okubo 2005, p. 13. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Schafer 1995, p. 5. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Okubo 2005, p. 18. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">McCrimmon 2004, p. 153. - McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 978-0-387-95447-9. MR 2014924. Zbl 1044.17001. Errata. <a href="https://books.google.com/books?isbn=9780387954479" target="_blank">https://books.google.com/books?isbn=9780387954479</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Schafer 1995, p. 28. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Okubo 2005, p. 16. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Schafer 1995, p. 30. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Schafer 1995, p. 128. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Okubo 2005, p. 17. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Knus et al. 1998, p. 451. - Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The book of involutions. Colloquium Publications. Vol. 44. With a preface by J. Tits. Providence, RI: American Mathematical Society. ISBN 0-8218-0904-0. Zbl 0955.16001. <a href="https://books.google.com/books?isbn=0821809040" target="_blank">https://books.google.com/books?isbn=0821809040</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Albert 1948a, p. 553. - Albert, A. Adrian (1948a). "Power-associative rings". Transactions of the American Mathematical Society. 64: 552–593. doi:10.2307/1990399. ISSN 0002-9947. JSTOR 1990399. MR 0027750. Zbl 0033.15402. <a href="https://doi.org/10.2307%2F1990399" target="_blank">https://doi.org/10.2307%2F1990399</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Rosenfeld 1997, p. 91. - Rosenfeld, Boris (1997). Geometry of Lie groups. Mathematics and its Applications. Vol. 393. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-4390-5. Zbl 0867.53002. <a href="https://books.google.com/books?isbn=0792343905" target="_blank">https://books.google.com/books?isbn=0792343905</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">It follows from the Artin's theorem. <a href="/wiki/Artin%27s_theorem" target="_blank">/wiki/Artin%27s_theorem</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">It follows from the Artin's theorem. <a href="/wiki/Artin%27s_theorem" target="_blank">/wiki/Artin%27s_theorem</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Jacobson 1968, p. 36. - Jacobson, Nathan (1968). Structure and representations of Jordan algebras. American Mathematical Society Colloquium Publications, Vol. XXXIX. Providence, R.I.: American Mathematical Society. ISBN 978-0-821-84640-7. MR 0251099. <a href="https://books.google.com/books?isbn=9780821846407" target="_blank">https://books.google.com/books?isbn=9780821846407</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Schafer 1995, p. 92. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Kokoris 1955, p. 710. - Kokoris, Louis A. (1955). "Power-associative rings of characteristic two". Proceedings of the American Mathematical Society. 6 (5). American Mathematical Society: 705–710. doi:10.2307/2032920. <a href="https://doi.org/10.2307%2F2032920" target="_blank">https://doi.org/10.2307%2F2032920</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Albert 1948b, p. 319. - Albert, A. Adrian (1948b). "On right alternative algebras". Annals of Mathematics. 50: 318–328. doi:10.2307/1969457. JSTOR 1969457. <a href="https://doi.org/10.2307%2F1969457" target="_blank">https://doi.org/10.2307%2F1969457</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Mikheev 1976, p. 179. - Mikheev, I.M. (1976). "Right nilpotency in right alternative rings". Siberian Mathematical Journal. 17 (1): 178–180. doi:10.1007/BF00969304. <a href="https://doi.org/10.1007%2FBF00969304" target="_blank">https://doi.org/10.1007%2FBF00969304</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Zhevlakov et al. 1982, p. 343. - Zhevlakov, Konstantin A.; Slin'ko, Arkadii M.; Shestakov, Ivan P.; Shirshov, Anatoly I. (1982) [1978]. Rings that are nearly associative. Translated by Smith, Harry F. ISBN 0-12-779850-1. <a href="https://www.researchgate.net/publication/260600596_Rings_that_are_nearly_associative" target="_blank">https://www.researchgate.net/publication/260600596_Rings_that_are_nearly_associative</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Schafer 1995, p. 148. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Bremner, Murakami & Shestakov 2013, p. 18. - Bremner, Murray; Murakami, Lúcia; Shestakov, Ivan (2013) [2006]. "Chapter 86: Nonassociative Algebras" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2nd ed.). CRC Press. ISBN 978-1-498-78560-0. <a href="https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf" target="_blank">https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Bremner, Murakami & Shestakov 2013, pp. 18–19, fact 6. - Bremner, Murray; Murakami, Lúcia; Shestakov, Ivan (2013) [2006]. "Chapter 86: Nonassociative Algebras" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2nd ed.). CRC Press. ISBN 978-1-498-78560-0. <a href="https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf" target="_blank">https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Albert 1948a, p. 554, lemma 4. - Albert, A. Adrian (1948a). "Power-associative rings". Transactions of the American Mathematical Society. 64: 552–593. doi:10.2307/1990399. ISSN 0002-9947. JSTOR 1990399. MR 0027750. Zbl 0033.15402. <a href="https://doi.org/10.2307%2F1990399" target="_blank">https://doi.org/10.2307%2F1990399</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Albert 1948a, p. 554, lemma 3. - Albert, A. Adrian (1948a). "Power-associative rings". Transactions of the American Mathematical Society. 64: 552–593. doi:10.2307/1990399. ISSN 0002-9947. JSTOR 1990399. MR 0027750. Zbl 0033.15402. <a href="https://doi.org/10.2307%2F1990399" target="_blank">https://doi.org/10.2307%2F1990399</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Schafer 1995, p. 14. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">McCrimmon 2004, p. 56. - McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 978-0-387-95447-9. MR 2014924. Zbl 1044.17001. Errata. <a href="https://books.google.com/books?isbn=9780387954479" target="_blank">https://books.google.com/books?isbn=9780387954479</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Okubo 2005, p. 13. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Rowen 2008, p. 321. - Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. American Mathematical Society. ISBN 0-8218-8408-5. <a href="https://books.google.com/books?isbn=0821884085" target="_blank">https://books.google.com/books?isbn=0821884085</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Kurosh 1947, pp. 237–262. - Kurosh, A.G. (1947). "Non-associative algebras and free products of algebras". Mat. Sbornik. 20 (62). MR 0020986. Zbl 0041.16803. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0020986" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0020986</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Schafer 1995, p. 4. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Okubo 2005, p. 24. - Okubo, Susumu (2005) [1995]. Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge University Press. doi:10.1017/CBO9780511524479. ISBN 0-521-01792-0. Zbl 0841.17001. <a href="https://books.google.com/books?isbn=0521017920" target="_blank">https://books.google.com/books?isbn=0521017920</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Schafer 1995, p. 14. - Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. Zbl 0145.25601. <a href="https://books.google.com/books?isbn=0486688135" target="_blank">https://books.google.com/books?isbn=0486688135</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Albert 2003, p. 113. - Albert, A. Adrian (2003) [1939]. Structure of algebras. American Mathematical Society Colloquium Publ. Vol. 24 (Corrected reprint of the revised 1961 ed.). New York: American Mathematical Society. ISBN 0-8218-1024-3. Zbl 0023.19901. <a href="https://books.google.com/books?isbn=0821810243" target="_blank">https://books.google.com/books?isbn=0821810243</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Knus et al. 1998, p. 451. - Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The book of involutions. Colloquium Publications. Vol. 44. With a preface by J. Tits. Providence, RI: American Mathematical Society. ISBN 0-8218-0904-0. Zbl 0955.16001. <a href="https://books.google.com/books?isbn=0821809040" target="_blank">https://books.google.com/books?isbn=0821809040</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">McCrimmon 2004, p. 57. - McCrimmon, Kevin (2004). A taste of Jordan algebras. Universitext. Berlin, New York: Springer-Verlag. doi:10.1007/b97489. ISBN 978-0-387-95447-9. MR 2014924. Zbl 1044.17001. Errata. <a href="https://books.google.com/books?isbn=9780387954479" target="_blank">https://books.google.com/books?isbn=9780387954479</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Koecher 1999, p. 57. - Koecher, Max (1999). Krieg, Aloys; Walcher, Sebastian (eds.). The Minnesota notes on Jordan algebras and their applications. Lecture Notes in Mathematics. Vol. 1710. Berlin: Springer-Verlag. ISBN 3-540-66360-6. Zbl 1072.17513. <a href="https://books.google.com/books?isbn=3540663606" target="_blank">https://books.google.com/books?isbn=3540663606</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
</ol>

Non-associative algebra open-in-new

Non-associative algebra