Why is  $O(n;k)$ not connected and has four connected components? Here $O(n;k) =\{A\in GL(n+k,\mathbb{C}) \mid A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$,

with $n$ instances of $1$, and $k$ instances of $-1$.

Why is $O(n;k)$ not connected and has four connected components when $nk\ge 1$? Here $O(n;k) =\{A\in GL(n+k,\mathbb{R}) \mid A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$,

with $n$ instances of $1$, and $k$ instances of $-1$.

Why is $O(n;k)$ not connected and has four connected components? $O(n;k) =\{A\in GL(n+k,\mathbb{C}) \mid A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$,

and it has n number 1, and k -1

Why is $O(n;k)$ not connected and has four connected components? Here $O(n;k) =\{A\in GL(n+k,\mathbb{C}) \mid A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$,

with $n$ instances of $1$, and $k$ instances of $-1$.

Why is O（n;k）not connected，and it has four connection branches？ $O（n；k）=\{A\in GL(n+k，C)|A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$，

and it has n number 1，and k -1

Why is $O(n;k)$ not connected and has four connected components? $O(n;k) =\{A\in GL(n+k,\mathbb{C}) \mid A^{T}GA=G\}$

where $G=\begin{pmatrix} 1&&&&&\\ &\ddots& & & &\\ &&1&& &\\ && &-1& &\\ && & &\ddots &\\ && & & &-1 \end{pmatrix}$,

and it has n number 1, and k -1