Anti-commutative algebra

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; Binary Lie algebra; Mal'tsev algebra).

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