View source for Binary Lie algebra

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''$BL$-algebra''

A linear algebra $A$ over a field $F$ any two elements of which generate a Lie subalgebra. The class of all binary Lie algebras over a given field $F$ generates a variety which, if the characteristic of $F$ is different from 2, is given by the system of identities

\begin{equation}x^2=J(xy,x,y)=0,\label{eq1}\end{equation}

where

$$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$

If the characteristic of $F$ is 2 and its cardinal number is not less than 4, the class of binary Lie algebras cannot be defined only by the system of identities \eqref{eq1}, also needed is the identity

$$J([(xy)y]x,x,y)=0.$$

The tangent algebra of an analytic local alternative [[loop]] is a binary Lie algebra and vice versa.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev, "Analytic loops"  ''Mat. Sb.'' , '''36 (78)''' :  3  (1955)  pp. 569–575  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.T. Gainov, "Binary Lie algebras of characteristic two"  ''Algebra and Logic'' , '''8''' :  5  (1969)  pp. 287–297  ''Algebra i Logika'' , '''8''' :  5  (1969)  pp. 505–522</TD></TR>
</table>