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We give a new construction of the Lie algebra of type $E_8$, in terms of$3\times3$ matrices, such that the Lie bracket has a natural description as thematrix commutator. This leads to a new interpretation of the Freudenthal-Titsmagic square of Lie algebras, acting on themselves by commutation.

An octonionic construction of $E_8$ and the Lie algebra magic square