It was proved by Dixmier

(http://archive.numdam.org/ARCHIVE/AIF/AIF_1957__7_/AIF_1957__7__315_0/AIF_1957__7__315_0.pdf)

that algebraic groups over the reals, are type I. Is a similar result known for algebraic groups over non-archimedean local fields (possibly of characteristic 0)? I am only aware of the result by Berstein

http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-P-tame-FAN.pdf

that reductive algebraic groups over non-archimedean local fields, are type I.