Maschke's theorem and the complex central-idempotent formula give central, pairwise orthogonal idempotents \(e_\chi \) acting on the indicated complex components. Coefficient conjugation sends \(e_\chi \) to \(e_{\overline \chi }\), so it fixes \(\epsilon _\Xi \); its coefficients are real. Average a Hermitian inner product on a complex representation over \(G\). Each group operator becomes unitary, whence \(\chi (g^{-1})=\overline {\chi (g)}\). The complex linear extension of \([g]\mapsto [g^{-1}]\) therefore sends \(e_\chi \) to \(e_{\overline \chi }\), and fixes \(\epsilon _\Xi \). This is coefficient conjugation and inversion separately; the latter restricts to the stated real involution.

For completeness, the regular character is \(r_G(g)=|G|\,\delta _{g,1}\) and decomposes as \(r_G=\sum _\chi \chi (1)\chi \) by [serre1977linear, Section 2.4, Proposition 5 and Corollary 1, p. 18]. Consequently \[\begin {aligned} \sum _\chi e_\chi &=\frac 1{|G|}\sum _{g\in G} \left (\sum _\chi \chi (1)\chi (g^{-1})\right )[g]\\ &=\frac 1{|G|}\sum _{g\in G}r_G(g^{-1})[g]=[1]. \end {aligned}\] Grouping by conjugation orbits preserves this sum. Applying \(\widehat \rho \) gives idempotence, mixed-product zero and sum \(I_H\). Orthogonality gives \(\rho (g)^*=\rho (g^{-1})\), so \(\widehat \rho (a^*)=\widehat \rho (a)^*\) for real coefficients. Thus each \(P_\Xi \) is self-adjoint; distinct ranges are perpendicular because \(\langle P_\Xi x,P_\Upsilon y\rangle =\langle x,P_\Xi P_\Upsilon y\rangle =0\). The sum identity gives existence of the decomposition, and applying each projector gives its uniqueness. Commutation with \(Q\) follows by linearity from commutation with every group operator.

To justify the real isotypic terminology, decompose \(H\) into irreducible real subrepresentations by Maschke. On any such \(U\), the real commuting idempotents \(P_\Xi \) have invariant ranges, so exactly one acts as the identity and the rest as zero. Its complexification therefore has constituents in just that orbit. Two real irreducibles assigned to the same orbit share a complex constituent: complex conjugation on each complexification exchanges the constituents indexed by \(\chi \) and \(\overline \chi \). Semisimplicity over \(\mathbb C\) then gives a nonzero complex intertwiner between their complexifications. In real bases, the real and imaginary parts of its matrix are real intertwiners, at least one nonzero. Its kernel and image are invariant, so irreducibility makes it an isomorphism. Conversely an isomorphism commutes with all the character projectors. This proves precisely the asserted grouping.