Could anyone point me to a reference for the following fact : 


Let $G$ denote the orthogonal or symplectic complex group and $H$ be a complex Lie group, then if $\rho\_1\,:H\, \rightarrow G$, $\rho\_2\,:H\, \rightarrow G$ are two equivalent representations (conjugated by the linear group), they are conjugated by the group $G$.



Equivalent orthogonal representations are orthogonaly equivalent
See Theorem 2.1 (b) in arXiv:math/0505268 [math.SG] for the symplectic case. The orthogonal case should be similar. 


