  We show vanishing of the second $L^p$-cohomology group for most semisimple
algebraic groups of rank at least 3 over local fields. More precisely, we show
this result for $\SL(4)$, for simple groups of rank $\geq 4$ that are not of
exceptional type or of type $D_4$ and for all semisimple, non-simple groups of
rank $\geq 3$. Our methods work for large values of $p$ in the real case and
for all $p>1$ in the non-Archimedean case. This result points towards a
positive answer to Gromov's question on vanishing of $L^p$-cohomology of
semisimple groups for all $p>1$ in degrees below the rank. The methods consist
in using a spectral sequence \`a la Bourdon-R\'emy, adapting a version of
Mautner's phenomenon from Cornulier-Tessera and concluding thanks to a
combinatorial case-by-case study of classical simple groups.
