Definition 1

An algebra g over K is called a Lie algebra over K if its multiplication (denoted by $(x, y) \mapsto [x, y]$) satisfies the identities:

(1)

(2)

for all $x, y, z$ in g.

The product $[x, y]$ is called the bracket of $x$ and $y$. Identity (2) is called the Jacobi identity.

The bracket $[x, y]$ is an alternating bilinear function of $x$ and $y$. We have the identity:

(3)

so that the Jacobi identity can be written:

(4)

Every subalgebra and every quotient algebra of a Lie algebra is a Lie algebra. Every product of Lie algebras is a Lie algebra. If $g$ is a Lie algebra, the opposite algebra $g^0$ is a Lie algebra and the mapping $x \mapsto -x$ an isomorphism of $g$ onto $g^0$, by virtue of identity (3).