Let $A$ be a central division algebra (of finite dimension) over a field $k$. Show that $[A,A] \neq A$.

Question

I am looking at the post A central division algebra is not its commutator and I have a few questions regarding the proof that was provided in the answer.

1. Why is $A$ a simple $k$-algebra?

My first observation is that $A$ is Artinian because it is finite-dimensional. Then, I normally think about the Jacobson Radical being trivial to arrive at $A$ being semisimple. Are all central division algebras simple? Every time I Google central division algebra, it leads me to the Wikipedia page for central simple algebras but there is no discussion on my question so I think I am misunderstanding something fundamental.

2. Given that $B =A \otimes_k \overline{k}$ how do I see that $[B,B] = [A,A] \otimes_k \overline{k}$? I usually view adding and subtracting pure tensors to be completely abstract.

Answer 1

As a hint for the first question, consider what a two sided ideal might look like in your division algebra, is there anything in the setup that forces this to be a trivial ideal? (Let $x$ be in this ideal…)

For the second part, imagine you had picked a basis over $k$. Can you use this to build a generating set of this commutator subalgebra over both $k$ and $\bar{k}$?