Let $D\subseteq GL_n(\mathbb{C})$ be the subgroup of diagonal matrices and let $T=D\cap U(n)$. Let us assume that $$ T\subseteq U(n) \subseteq GL_n(\mathbb{C})$$ is a maximal torus and a maximal compact subgroup for $GL_n(\mathbb{C})$.
Question: Assuming this, what is the easiest way to show that $$T\cap SL_n(\mathbb{C})\subseteq U(n)\cap SL_n(\mathbb{C}) \subseteq SL_n(\mathbb{C})$$ is a maximal torus and maximal compact subgroup for $SL_n(\mathbb{C})$? I would prefer to use elementary group theory and point set topology, and to avoid Lie algebras.