Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$.
We have $V \cong \bigoplus_{i=1}^mW_i^{n_i}$. The projection of $V$ onto its isotypic component $W_i^{n_i}$ is given by $$p_i=\frac{\dim(W_i)}{\dim(\text{End}_G(W_i))|G|}\sum_{i}\chi_i(g^{-1})\rho_g$$
This formula is both for the case when the ground field is $\mathbb{C}$, in which case $\dim(\text{End}_G(W_i))=1$, or the ground field is $\mathbb{R}$, in which case $\dim(\text{End}_G(W_i))=1,2, \text{or}\ 4$.
Moreover, when the field is $\mathbb{C}$, each isotypic component can be identified as $W_i^{n_i}\cong H_i^{\dim(W_i)}$ where $H_i \cong \text{Hom}_G(W_i,W_i^{n_i})$. In some textbooks, I have seen if matrix $R_i$ is any unitary realization of $W_i$ then $$p_i(j,j)=\frac{\dim(W_i)}{|G|}\sum_{i}[R_i(g^{-1})]_{jj}\rho_g$$ is an unitary projection onto $H_i$ for any $1 \le j \le n_i$. Here $[R_i(g^{-1})]_{jj}$ is the $(j,j)$ elements of $R_i(g^{-1})$.
Now, let's talk about the real case. In this case, $\dim(H_i)=n_i \dim(\text{End}_G(W_i))$. So we cannot identify $W_i^{n_i}$ with $H_i^{\dim(W_i)}$ unless $\dim(\text{End}_G(W_i)$ is one ($W_i$ is called a real-type irreducible representation over $\mathbb{R}$).
My questions:
What can we say about $$p_i(j,j)=\frac{\dim(W_i)}{\dim(\text{End}_G(W_i))|G|}\sum_{i}[R_i(g^{-1})]_{jj}\rho_g,$$ where $R_i$ is any orthogonal realization of $W_i$ and $W_i$ is not real-type?
Can we say $W_i^{n_i} \otimes \text{End}_G(W_i) \cong H_i^{\dim(W_i)}$ for all cases? And if the answer is yes, is $p_i(j,j)$ is a projection of $W_i^{n_i} \otimes \text{End}_G(W_i)$ onto $H_i$?
I construct $p_i(j,j)$ and I saw $\dim(\text{image}(p_i(j,j)))=n_i \dim(\text{End}_G(W_i))= \dim(H_i)$ which seems fine but $p_i(j,j)^2 \ne p_i(j,j)$. What is the problem?
Do you know any book that talked about the decomposition of real representations their projections or the central idempotent of real division algebras in detail?
