Example 2

In Example 1 choose $L$ to be the associative algebra of endomorphisms of a $K$-module $E$. We obtain the Lie algebra of endomorphisms of $E$, denoted by $gl(E)$. (If $E = K^n$, the Lie algebra $gl(E)$ is denoted by $gl(n, K)$.)

Every Lie subalgebra of $gl(E)$ is a Lie algebra over $K$. In particular: (1) If $E$ is given a (not necessarily associative) algebra structure, the derivations of $E$ form a Lie algebra over $K$. (2) If $E$ admits a finite basis, the endomorphisms of $E$ of zero trace form a Lie algebra over $K$ denoted by $sl(E)$ (or $sl(n, K)$ if $E = K^n$).