Characterization of identity component of $O(p,q)$

Question

For $p+q=n$ let $O(p,q)$ be the group of matrices $P\in GL_n(\mathbb{R})$ such that $PI_{(p,q)}P^t = I_{(p,q)}$ with $$ I_{(p,q)} = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}. $$ I know that this group has four connected components. I want to prove the following equality that characterize the component containing the identity which, I saw, is called $SO^+(p,q)$ with $$ SO^+(p,q) = \left\{ \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in SO(p,q) \mid \det A > 0 \right\} $$ such that $A$ is a $p \times p$ matrix and $D$ is a $q \times q$ matrix. I also know, by computing with block decomposition, that the condition of being in $O(p,q)$ implies that $|\det A| \geq 1$ and $|\det D| \geq 1$.