This paper is based on a theorem by G. A. Margulis allowing to decide when a lattice \(\Gamma\) in a semisimple Lie group \(G\) is arithmetic (supposing that \(G\) is connected, adjoint and without compact factors). The lattice \(\Gamma\) is a discrete subgroup of \(G\) such that \(\Gamma \in G\) has \(G\)-invariant finite volume. The concept of arithmeticity involves that of commensurability. Two subgroups \(G_1\) and \(G_2\) of \(G\) are commensurate if their intersection \(G_1 \cap G_2\) is of finite index in \(G_1\) and in \(G_2\). Commensurability is an equivalence relation. Considering now an algebraic semisimple group \(H\) defined on \(Q\), the group \(H_R\) of points with real coordinates and its subgroup \(H_Z\) of points with integral coordinates, then \(H_Z\) is a lattice in \(H_R\). Roughly speaking, the lattice \(\Gamma\) is arithmetic if it is commensurate with a properly defined homomorphic image of \(H_Z\) in \(G\). The point is that not all lattices \(\Gamma\) in \(G\) are arithmetic. One considers in \(G\) subgroups which are commensurate and conjugated to \(\Gamma\) and one defines the commensurator of \(\Gamma\) in \(G\) as the group given by \(\text{Comm}_G \Gamma = \{g \in G \mid g\Gamma g^{- 1} \sim \Gamma\}\) where “\(\sim\)” indicates commensurate equivalence. The theorem by Margulis says that a necessary and sufficient condition for \(\Gamma\) to be arithmetic is that the commensurator \(\text{Comm}_G \Gamma\) is dense in \(G\). The paper, written in honor of Armand Borel, analyzes this theorem, gives a proof and a canonical construction of the arithmeticity relation and formulates a number of alternative properties implying the arithmeticity of the lattice \(\Gamma\).