  Let $\Gamma \leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ be a
group acting freely and transitively on the product of two regular trees of
degree $d_1$ and $d_2$. We develop an algorithm which computes the closure of
the projection of $\Gamma$ on $\mathrm{Aut}(T_{d_t})$ under the hypothesis that
$d_t \geq 6$ is even and that the local action of $\Gamma$ on $T_{d_t}$
contains $\mathrm{Alt}(d_t)$. We show that if $\Gamma$ is torsion-free and $d_1
= d_2 = 6$, exactly seven closed subgroups of $\mathrm{Aut}(T_6)$ arise in this
way. We also construct two new infinite families of virtually simple lattices
in $\mathrm{Aut}(T_{6}) \times \mathrm{Aut}(T_{4n})$ and in
$\mathrm{Aut}(T_{2n}) \times \mathrm{Aut}(T_{2n+1})$ respectively, for all $n
\geq 2$. In particular we provide an explicit presentation of a torsion-free
infinite simple group on $5$ generators and $10$ relations, that splits as an
amalgamated free product of two copies of $F_3$ over $F_{11}$. We include
information arising from computer-assisted exhaustive searches of lattices in
products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace,
some of our results are used to show that abstract and relative commensurator
groups of free groups are almost simple, providing partial answers to questions
of Lubotzky and Lubotzky-Mozes-Zimmer.
