Lie subalgebra

Summary

A Lie subalgebra of a Lie algebra $\mathfrak g$ is a linear subspace that is closed under the Lie bracket. With the restricted bracket it is a Lie algebra in its own right. Subalgebras of the matrix Lie algebra $\mathfrak{gl}_n$ give most of the classical examples. For a Lie group, the connected Lie subgroups correspond to the Lie subalgebras of its Lie algebra.