Showing that $O(a)\times O(b)$ is maximal compact in $O(a,b)$

Question

Let $O(a,b)=\{A\in Gl(n,\mathbb{C}) |\ A^TI_{a,b}A=I\}$ where $a+b=n$ and $I_{a,b}=\begin{bmatrix}I_a & 0 \\ 0 & -I_b \end{bmatrix}$. I want to determine the maximal compact subgroup of $O(a,b)$. I've found online that this is the group $O(a)\times O(b)=\{\begin{bmatrix}A & 0 \\ 0 & B \end{bmatrix}\ |\ A\in O(a),B\in O(b)\}$. I've managed to show that $O(a)\times O(b)$ it is a compact subgroup, however I'm having trouble with showing that it's maximal. Any hint is appriciated.