Why is the commutator subalgebra of a Lie algebra a linear subspace?

Question

One defines commutator subalgebra of Lie algebra $\mathfrak{g}$ as $[\mathfrak{g},\mathfrak{g}]$. Why is it really subalgebra:

Why $\forall_{a,b,c,d \in \mathfrak{g}} \exists_{e,f \in \mathfrak{g}} [a,b] + [c,d]=[e,f]$ ?

Answer 1

What Tobias said. It's not true that $[a,b]+[c,d]$ necessarily equals some $[e,f]$, but we consider all linear combinations of basic commutators as being in $[\mathfrak g,\mathfrak g]$. It's similar to how the commutator subgroup of a group is the subgroup generated by all commutators, not just the commutators themsleves.