We define a symmetric bilinear form $[~]_{n+k}$ on $\mathbb{R}^{n+k}$ by the formula $$ [x, y]_{n+k} = x_1y_1 + x_2y_2 + x_3y_3 + \cdots \cdots + x_ny_n - x_{n+1}y_{n+1} - x_{n+2}y_{n+2} - \cdots \cdots - x_{n+k}y_{n+k} $$
where $n, k \in \mathbb{Z}^+$ and $x,y \in \mathbb{R}^{n+k}$.
The set of $(n+k) \times (n+k)$ real matrices $A$ which preserve this form; i.e., $[Ax, Ay]_{n+k} = [x, y]_{n+k}$ for all $x,y \in \mathbb{R}^{n+k}$ forms a group and this is called the generalized orthogonal group, $O(n;k)$.
How can I prove that $O(n; k)$ and $O(k; n)$ are the same group?
