Let $V$ be a real vector space and $G$ be a finite group and $\rho:G \rightarrow GL(V)$ be an irreducible representation over $\mathbb{R}$. It is of quaternionic type if $\text{Hom}_G(\rho,\rho)$, the set of commuting linear maps, is isomorphic to $\mathbb{H}$ as an $\mathbb{R}$-algebra. Is there a representation $\rho'$ equivalent to $\rho$ whose matrix expression is in $H_k$ for some $k$?
Here $H_k$ is the subset of $\mathbb{R}^{4k \times 4k}$ whose each $4 \times 4$ subblock is of the form \begin{pmatrix} a&-b &-c &-d \\ b&a &-d &c \\ c&d &a &-b \\ d&-c &b &a \\ \end{pmatrix} for some $a,b,c,d \in \mathbb{R}$.
Motivation:
I am interested in the real irreducile representation and wish to prove similar result to Schur orthogonality relation. (I.e. counterpart to ``coordinates statement'' part of https://en.wikipedia.org/wiki/Schur_orthogonality_relations )
For quaternionic type irreducible representation, if the answer to my question is yes, I think I can prove the orthogonlaity.