Let Gamma <= Aut(T-d1) x Aut(T-d2) 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 that computes the closure of the projection of Gamma on Aut(T-dt) under the hypothesis that dt >= 6 is even and that the local action of Gamma on T-dt contains Alt(d(t)). We show that if Gamma is torsion-free and d(1) = d(2)= 6, exactly seven closed subgroups of Aut(T-6) arise in this way. We also construct two new inunite families of virtually simple lattices in Aut(T-6)xAut(T-4n) and in Aut(T-2n) xAut(T2n+1), respectively, for all n >= 2. In particular, we provide an explicit presentation of a torsion-free inunite simple group on thorn generators and E 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.
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