Except for the case $G=\SL_2$, worked out in a previous paper by the first
author and A. Henke, very little is known about the structure of the
indecomposable direct summands of a tensor product of two simple modules of
restricted highest weight, for a given semisimple, simply-connected, linear
algebraic group $G$ over an algebraically closed field in positive
characteristic. This paper studies the problem for the case $G=\SL_3$ in
characteristics 2 and 3, obtaining along the way the submodule structure of
various Weyl and tilting modules.