In David Applebaum's "Probability on Compact Lie Groups", ch.2 page 36, we have the following definition of the non-commutative Fourier transform where $G$ is e.g. a compact Lie group, $f \in L^1(G; \mathbb{C})$ and $\pi$ is a complex unitary irrep of $G$:
$$ \widehat{f}(\pi) := \int_G \pi(g^{-1}) f(g) dg $$
Question: Analogous to how we may define the sine and cosine series instead of the complex Fourier series for a periodic function on $\mathbb{R}$, is there a definition of non-commutative Fourier transform (on compact Lie groups is sufficient) in terms of only real-valued functions and representations on real vector spaces?
If it simplifies things, I really only care about the case $G=SO(3)$ and real-valued functions $L^1(SO(3); \mathbb{R})$.
Thanks for any assistance you can provide :)