Lie subalgebra

Remarks

Example: in $\mathfrak{gl}_n(k)$, the space $\mathfrak b_n$ of upper triangular matrices is a subalgebra. If $A$ and $B$ are upper triangular, so are $AB$ and $BA$, hence so is $[A, B] = AB - BA$.

Non-example: the symmetric matrices in $\mathfrak{gl}_2(\R)$ form a subspace but not a subalgebra. For $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, the bracket is $[A, B] = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, which is not symmetric.

A subalgebra $\mathfrak h$ with the restricted bracket is itself a Lie algebra. The intersection of subalgebras is a subalgebra, so each subset generates a smallest subalgebra containing it.