What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

Question

I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. Does anyone know where I could find such a formula?

Answer 1

Every complex irreducible representation of $D_{2n}$ can be defined over $\mathbb{R}$, so it's just the same as over $\mathbb{C}$:

Two copies of $\mathbb{R}$ and $(n-1)/2$ copies of $M_2(\mathbb{R})$ for $n$ odd.

Four copies of $\mathbb{R}$ and $(n-2)/2$ copies of $M_2(\mathbb{R})$ for $n$ even.