  We are motivated by the question that for which class of right-angled Artin
groups (RAAG's), the quasi-isometry classification coincides with
commensurability classification. This is previously known for RAAG's with
finite outer automorphism groups. In this paper, we identify two classes of
RAAG's, where their outer automorphism groups are allowed to contain adjacent
transvections and partial conjugations, hence infinite. If $G$ belongs to one
of these classes, then any other RAAG $G'$ is quasi-isometric to $G$ if and
only if $G'$ is commensurable to $G$. We also show that in this case, there
exists an algorithm to determine whether two RAAG's are quasi-isometric by
looking at their defining graphs. Compared to the finite out case, as well as
the previous quasi-isometry rigidity results for symmetric spaces, thick
Euclidean buildings and mapping class groups, the main issue we need to deal
with here is the reconstruction map may not have nice properties as before, or
may not even exist. We introduce a deformation argument, as well as techniques
from cubulation to deal with this issue.
