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I need to understand the concept of a derived subalgebra. What does this mean in the most basic terms and how does one prove that an algebra is the derived subalgebra of another?

For example, what would one need to do to show that $\mathfrak{sl}(V)$ is a derived subalgebra of $\mathfrak{gl}(V)$?

Artyniusz
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The derived subalgebra of a Lie algebra $L$ is the subspace of $L$ generated by all of the elements of the form $[x,y]$ with $x$ and $y$ in $L$.

Mariano Suárez-Álvarez
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