Let $G = \langle\sigma\rangle$ be the cyclic group of order $4$ and consider the group algebra $\mathbb RG$. By Maschke's theorem, $\mathbb RG$ is semisimple and so by Wedderburn's theorem, it is a product of matrix rings over some division rings.
For a tutorial problem, I'm required to determine the Wedderburn components. Since $\mathbb R$ is not algebraically closed, I don't know how to proceed. I think, based on the proof of Wedderburn's theorem, I need to start by finding the decomposition of $\mathbb RG$ into a sum of simple rings, but I don't even know how to do that. I know $\mathbb RG \cong \mathbb R \oplus \mathbb R\sigma \oplus \mathbb R\sigma^2 \oplus \mathbb R\sigma^3$ as $\mathbb R$-spaces but those summands aren't ideals of $\mathbb RG$ so I don't know where to go from here.