Split-quaternion

Split-quaternion multiplication<table><tbody><tr><th>×</th><th>1</th><th>i</th><th>j</th><th>k</th></tr><tr><th>1</th><td>1</td><td>i</td><td>j</td><td>k</td></tr><tr><th>i</th><td>i</td><td>−1</td><td>k</td><td>−j</td></tr><tr><th>j</th><td>j</td><td>−k</td><td>1</td><td>−i</td></tr><tr><th>k</th><td>k</td><td>j</td><td>i</td><td>1</td></tr></tbody></table>
<p>In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, the split-quaternions  or coquaternions form an <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> introduced by <a href="/facts/James_Cockle_(lawyer)/5MFqw4T0">James Cockle</a> in 1849 under the latter name. They form an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> of dimension four over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.
</p><p>After introduction in the 20th century of coordinate-free definitions of <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> and <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebras</a>, it was proved that the algebra of split-quaternions is <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a> to the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of the 2×2 <a href="/facts/Real_matrix/qa8FI4ko">real matrices</a>. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries.
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Split-quaternion open-in-new

Split-quaternion