  Let $X$ be a normal projective variety of dimension $n$ and $G$ an abelian
group of automorphisms such that all elements of $G\setminus \{\mathrm{id}\}$
are of positive entropy. Dinh and Sibony showed that $G$ is actually free
abelian of rank $\le n - 1$. The maximal rank case has been well understood by
De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\mathrm{rank}
G = n - 2$.
