Representations of infinite dimension orthogonal groups of quadratic forms with finite index

We study representations $G\to H$ where $G$ is either a simple Lie group with real rank at least 2 or an infinite dimensional orthogonal group of some quadratic form of finite index at least 2 and $H$ is such an orthogonal group as well. The real, complex and quaternionic cases are considered. Contrarily to the rank one case, we show that there is no exotic such representations and we classify these representations. On the way, we make a detour and prove that the projective orthogonal groups $\mathop{PO}_\mathbf{K}(p,\infty)$ or their orthochronous component (where $\mathbf{K}$ denotes the real, complex or quaternionic numbers) are Polish groups that are topologically simple but not abstractly simple.