  Let $\mathbf{G}$ be an algebraic group over a local field $\mathbf k$ of
characteristic zero. We show that the locally compact group $\mathbf G(\mathbf
k)$ consisting of the $\mathbf k$-rational points of $\mathbf G$ is of type I.
Moreover, we complete Lipsman's characterization of the groups $\mathbf G$ for
which every irreducible unitary representation of $\mathbf G(\mathbf k)$ is a
CCR representation and show at the same time that such groups $\mathbf
G(\mathbf k)$ are trace class as studied recently by Deitmar and van Dijk.
