  There is a product decomposition of a compact connected Lie group $G$ at the
prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in
homology. Then in studying the multiplicative structure of the $p$-localization
of $G$, the Samelson products of the factor space inclusions of the mod $p$
decomposition are fundamental. This paper determines (non-)triviality of these
fundamental Samelson products in the $p$-localized exceptional Lie groups when
the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular.
