There is indeed a slight problem on p.91 of Bekka-Mayer, as they state that ${\cal H}_{\pi_I}$ is ``the subspace of vectors which are fixed by all $G_i,\;i\in I$, but not by any of the others.''
This condition clearly does not define a linear subspace. That problem may be overcome as follows: let $\pi=\int^\oplus_X \rho_x d\mu(x)$ be a direct integral decomposition of $\pi$ into irreducible representations $\rho_x$, over some measure space $(X,\mu)$. Since simple Lie groups are type I, we may write $\rho_x=\rho_{1,x}\otimes...\otimes \rho_{n,x}$, where $\rho_{i,x}$ is an irreducible representation of $G_i$. For $I\subset\{1,...,n\}$, define a subset $X_I$ of $X$, as the set of $x$'s in $X$ such that $\rho_{i,x}$ is trivial for $i\in I$, and non-trivial for $i\notin I$. Then the $X_I$'s, for $I$ running over all subsets of $\{1,...,n\}$, is a measurable partition of $X$; defining $\pi_I=:\int^\oplus_{X_I}\rho_x d\mu(x)$, we get $\pi=\bigoplus_I \pi_I$ as desired.