On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a discrete nontrivial normal subgroup (essentially $\mathbb{Z}$) does not have a faithful finite-dimensional representation. They denote this quotient group by $H$.
It starts by saying,
"One way to see this is to argue that in any irreducible finite-dimensional representation $V$ the center $S^1$ of $H$, being compact and abelian, must be diagonalizable."
This doesn't seem right. Rotations of the plane are quite visibly not diagonalizable over $\mathbb{R}$. Why does "compact and abelian" imply "diagonalizable"?