Special linear group is perfect

Statement
Let $$k$$ be any field. Consider the special linear group $$SL_n(k)$$: the group of $$n \times n$$ matrices over $$k$$ that have determinant $$1$$. Then, the following are true:


 * For $$n \ge 3$$, $$SL_n(k)$$ is a perfect group for any field $$k$$.
 * For $$n = 2$$, $$SL_n(k)$$ is a perfect group if $$k$$ has more than three elements.

Related facts

 * Derived subgroup of general linear group is special linear group
 * Symplectic group is perfect
 * Derived subgroup of orthogonal group is special orthogonal group

Facts used

 * 1) uses::Elementary matrices generate the special linear group
 * 2) uses::Every elementary matrix is a commutator of unimodular matrices: If $$n \ge 3$$ or $$n = 2$$ and $$k$$ has more than three elements, every elementary $$n \times n$$ matrix is a commutator of two elements of $$SL_n(k)$$.

Proof
The proof follows by piecing together facts (1) and (2).