Leibniz algebra

Remarks

The identity says that each left multiplication $b \mapsto [a, b]$ is a derivation of the product. A right Leibniz algebra satisfies instead $[\,[a, b], c] = [a, [b, c]] + [\,[a, c], b]$, saying that each right multiplication is a derivation; the opposite product turns one kind into the other.

Example: every Lie algebra is a Leibniz algebra, since by antisymmetry the Jacobi identity can be rewritten as $[a, [b, c]] = [\,[a, b], c] + [b, [a, c]]$. A Leibniz algebra that is not a Lie algebra: $L = k^2$ with basis $x, y$, $[x, x] = y$ and all other products of basis vectors zero. Every product lies in $k y$ and every product with $y$ is zero, so all three terms of the identity vanish, while $[x, x] \neq 0$.

Non-example: $L = k$ with basis $x$ and $[x, x] = x$. Then $[x, [x, x]] = x$ while $[\,[x, x], x] + [x, [x, x]] = 2x$, and these differ because $x \neq 0$.

A Leibniz algebra is a Lie algebra if and only if $[x, x] = 0$ for all $x$. The span of all squares $[x, x]$ is a two-sided ideal, and the quotient by it is a Lie algebra.