Abelian Lie algebra

Remarks

Example: the diagonal matrices in $\mathfrak{gl}_n(k)$ form an abelian Lie subalgebra. Two diagonal matrices $D, D'$ satisfy $DD' = D'D$, so $[D, D'] = 0$.

Non-example: $\mathfrak{sl}_2(k)$ with $\operatorname{char} k \ne 2$ is not abelian. For $h = \operatorname{diag}(1, -1)$ and $e = E_{12}$, one has $[h, e] = 2e \ne 0$.

Equivalent forms: $\mathfrak g$ is abelian if and only if its center is $\mathfrak g$, if and only if $[\mathfrak g, \mathfrak g] = 0$. Every vector space $V$ with the zero bracket is an abelian Lie algebra, and every linear map between abelian Lie algebras is a homomorphism.