"I will begin by reviewing the unified description of pure Super Yang-Mills (SYM) Theory (consisting of just a gauge field and gaugino) in dimensions 3, 4, 6, and 10 over the four normed division algebras R, C, H, and O. Dimensionally reducing these initial theories into dimensions 3, 4, 5, 6, 7, 8, 9, 10 gives a plethora of SYM theories written over the division algebras, with a single master Lagrangian to rule them all. In particular, in D = 3 spacetime dimensions, the SYM theories with N = 1, 2, 4, and 8 supersymmetries enjoy a unified description over R, C, H, and O, respectively. In each spacetime dimension, maximally supersymmetric theories are written over the octonions.
In apparently completely different developments, a popular thread in attempts to understand the quantum theory of gravity is the idea of "gravity as the square of Yang-Mills". The idea in its most basic form is that a symmetric tensor (graviton) can be built from the symmetric tensor product of two vectors (Yang-Mills fields), an idea which can be extended to obtain entire supergravity multiplets from tensor products of SYM multiplets. Having established a division-algebraic description of Super Yang-Mills theories, I will then demonstrate how tensoring these multiplets together results in supergravity theories valued over tensor products of division algebras.
In D = 3, there are 4 SYM theories (N = 1, 2, 4, 8 over R, C, H, O) and so there are 4 x 4 = 16 possible supergravity theories to obtain by "squaring Yang-Mills". The global symmetries of these 16 division-algebraic SYM-squared supergravity theories are precisely those belonging to the 4 x 4 Freudenthal-Rosenfeld-Tits "magic square" of Lie algebras! Furthermore, the scalar fields in these supergravity theories describe non-linear sigma models, whose target space manifolds are division algebraic projective planes! Performing the same tensoring of SYM theories in spacetime dimensions D > 3 results in a whole "magic pyramid" of supergravities, with the magic square at the base in D = 3 and Type II supergravity at the apex in D = 10. This construction gives an explicit octonionic explanation of many of the mysterious appearances of exceptional groups within string/M-theory and supergravity."