= Solution
Let $G=C_3=\langle g\rangle$ and let $e_0,e_1,e_2$ be the standard basis of its real <regular representation>, with $ge_j=e_{j+1}$ modulo three. Set
$$
U=\operatorname{span}\{e_0+e_1+e_2\},
\qquad
W=\{x_0e_0+x_1e_1+x_2e_2:x_0+x_1+x_2=0\}.
$$
Both are invariant, they are orthogonal, and
$$
\boxed{\mathbb RG=U\oplus W}.
$$
The line $U$ is the <trivial representation>. On the plane $W$, the generator acts as a rotation through $2\pi/3$. Such a rotation has no real eigenline, so $W$ has no nonzero proper invariant real subspace and is irreducible.
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