Consider as an example $$O(2)=\left\{\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\middle|\ \ \theta\in \Bbb R\right\},$$ clearly for $g\in O(2)$ one has $g_{11}-g_{22}=0=g_{12}+g_{21}$, ie we have two non-trivial linear relations on the components.
$O(2)$ is however quite the black sheep in the family $O(p,q)$. I am especially interested in $O(2,1)$, my question is:
On $O(2,1)$ does there exist any non-trivial linear relation between components?
So far I let a software evaluate the matrix exponential of a general element of $\mathfrak{so}(2,1)$ and I get a really bad expression. It looks like as if there are no such linear relations in this case, but just because I cannot find one doesn't mean none exist.
A generalisation of this question would be whether such relations exist on $O(p,q)$ for $p+q>2$. This is also interesting, but right now I am most concerned about the special case.