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Let $\Gamma \leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ be agroup acting freely and transitively on the product of two regular trees ofdegree $d_1$ and $d_2$. We develop an algorithm which computes the closure ofthe 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 thisway. We also construct two new infinite families of virtually simple latticesin $\mathrm{Aut}(T_{6}) \times \mathrm{Aut}(T_{4n})$ 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-freeinfinite simple group on $5$ generators and $10$ relations, that splits as anamalgamated free product of two copies of $F_3$ over $F_{11}$. We includeinformation arising from computer-assisted exhaustive searches of lattices inproducts of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace,some of our results are used to show that abstract and relative commensuratorgroups of free groups are almost simple, providing partial answers to questionsof Lubotzky and Lubotzky-Mozes-Zimmer.

New simple lattices in products of trees and their projections