Consider the real Lie algebra $\mathfrak{so}_{p,q}$, is there a way to classify the irreducible representations of this algebra in $\mathbb{R}^{p+q}$?
For example the $ad$ representation is given by,
$$ [\Delta_{i,j} ; Y_k] = \left\{\begin{array}{cl} -Y_j & \text{if}\ k = i \\ \frac{\alpha_i}{\alpha_j}Y_i &\text{if}\ k=j \end{array}\right. $$
where $\Delta_{i,j}$ is the standart basis for $\mathfrak{so}_{p,q}$ (if needed I can give it in term of elementary matrices), $\alpha_i=1$ when $1\leq i\leq p$ and $\alpha_i=-1$ when $p+1\leq i\leq p+q$ and $(Y_1,\cdots,Y_{p+q})$ is a basis of $\mathbb{R}^{p+q}$.
It is an irreducible representation, does any non equivalent irreducible representation exists ?
Thank you!