Lower central series of general linear group stabilizes at special linear group

Statement
Let $$k$$ be a field and $$n$$ be a natural number. Then, the lower central series of $$Gl_n(k)$$ stabilizes in one step at $$Sl_n(k)$$. In other words, it looks like:

$$GL_n(k) \ge SL_n(k) \ge SL_n(k) \ge SL_n(k) \ge \dots$$.

This holds under either of these conditions:


 * $$n \ge 3$$.
 * $$k$$ has at least three elements.

Note that for $$n \ge 3$$ and $$k$$ having exactly two elements, $$GL_n(k) = SL_n(k)$$, so the stabilization occurs at the first step itself. Also, for $$n = 1$$, the result holds trivially, since $$GL_1(k)$$ is Abelian and $$SL_1(k)$$ is trivial.

Related facts

 * Special linear group is perfect: This holds under slightly more restrictive circumstances: $$k$$ should have more than three elements.
 * Commutator subgroup of general linear group is special linear group: This is a weaker version, and holds under the same circumstances.

Facts used

 * 1) uses::Every elementary matrix is the commutator of an invertible and an elementary matrix: This holds under the same hypotheses: $$n \ge 3$$ or $$k$$ has at least three elements.
 * 2) uses::Elementary matrices generate the special linear group

The commutator subgroup is the special linear group
By facts (1) and (2), the commutator subgroup of $$GL_n(k)$$ contains $$SL_n(k)$$. On the other hand, $$Sl_n(k)$$ is the kernel of a homomorphism from $$Gl_n(k)$$ to the multiplicative group of $$k$$, which is Abelian. Thus, the commutator subgroup of $$Gl_n(k)$$ is contained in $$Sl_n(k)$$. Combining these facts, we get that the commutator subgroup of $$Gl_n(k)$$ equals $$SL_n(k)$$.

Further members of the lower central series
By fact (1), every elementary matrix can be expressed as the commutator between an element of $$GL_n(k)$$ and an element of $$SL_n(k)$$. Fact (2) thus yields:

$$SL_n(k) \le [GL_n(k), SL_n(k)]$$.

On the other hand, we know that the right side is contained in $$SL_n(k)$$, since $$SL_n(k) = [GL_n(k),GL_n(k)]$$. Thus, we get:

$$SL_n(k) = [GL_n(k), GL_n(k)]$$.

Thus, the lower central series stabilizes at $$SL_n(k)$$.