  We study the structure of bounded simple weight $\mathfrak{sl}(\infty)$-,
$\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, which have been
recently classified in [6]. Given a splitting parabolic subalgebra
$\mathfrak{p}$ of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$,
$\mathfrak{sp}(\infty)$, we introduce the concepts of $\mathfrak{p}$-aligned
and pseudo $\mathfrak{p}$-aligned $\mathfrak{sl}(\infty)$-,
$\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, and give necessary
and sufficient conditions for bounded simple weight modules to be
$\mathfrak{p}$-aligned or pseudo $\mathfrak{p}$-aligned. The existence of
pseudo $\mathfrak{p}$-aligned modules is a consequence of the fact that the Lie
algebras considered have infinite rank.
