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 $-1$.