There is a close relationship between
the four normed division algebras
The structure of the normed division algebras governs the existence of super-∞-Lie algebras such as the supergravity Lie 3-algebra. By the D'Auria-Fre formulation of supergravity the ∞-Lie algebra valued forms with values in these constitute the field content of (11-dimensional) supergravity.
Normed division algebras are used to construct Lie 2- and 3-superalgebras extending the Poincaré Lie superalgebra, and then Lie 2- and 3-supergroups extending the Poincaré Lie supergroups, here:
Rafael I. Nepomechie, Nonabelian bosonization, triality, and superstring theory Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210
I. Bars, D. Nemschansky and S. Yankielowicz, SLACPub- 3758.
H. Tachibana, K. Imeda, Octonions, superstrings and ten-dimensional spinors , Il nuovo cimento, Vol 104 B N.1
The relation of the division algebras to ordinary (Lie algebraic) extensions of the super Poincare Lie algebra is discussed in