The Cartan-Brauer-Hua theorem states that
Let $K\subset D$ be division rings so that whenever $x\in D$ is a nonzero element, $xKx^{-1}\subset K$. Show that either $K\subset Z(D)$ or $K=D$.
This theorem is partially named after Hua due to his 1-liner proof:
If $ab\neq ba$, then $$a=(b^{-1}-(a-1)^{-1}b^{-1}(a-1))(a^{-1}b^{-1}a-(a-1)^{-1}b^{-1}(a-1))^{-1}$$
It is all very nice, but the point is, how did all these come up? It is not obvious even if it has been written out, but how the h*ll was this constructed? What was the intuition?
