View source for Anti-commutative algebra

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A linear algebra over a field in which the identity

\begin{equation}x^2=0\label{*}\end{equation}

is valid. If the characteristic of the field differs from 2, the identity \eqref{*} is equivalent with the identity $xy=-yx$. All subalgebras of a free anti-commutative algebra are free. The most important varieties of anti-commutative algebras are Lie algebras, Mal'tsev algebras and binary Lie algebras (cf. [[Lie algebra|Lie algebra]]; [[Binary Lie algebra|Binary Lie algebra]]; [[Mal'tsev algebra|Mal'tsev algebra]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Shirshov,   "Subalgebras of free commutative and free anti-commutative algebras"  ''Mat. Sb.'' , '''34 (76)''' :  1  (1954)  pp. 81–88  (In Russian)</TD></TR></table>