check that $x,y,z$ span a 3-dimensional Lie subalgebra $L$ of $\mathbb{gl}(V)$

Question

Suppose $V$ is a 3-D vector space over a field $k$ with basis $B=\{v_1,v_2, v_3\}$ and consider the linear operators $x,y,z\in\mathbb{gl}(V)$ whose matrices with respect to $B$ are some 3 by 3 $X, Y$ and $Z$. How would one go about verifying $[x,y],[x,z],[y,z]$ are linear combinations of $x,y,z$ and finding these combinations (i.e check that $x,y,z$ span a 3-dimensional Lie subalgebra $L$ of $\mathbb{gl}(V)$)?