A geometric construction of the exceptional Lie algebras \(F_4\) and \(E_8\) (Q958160)

In this nice note the author presents a remarkable geometric construction of the rank four and eight exceptional Lie algebras \(F_4\) and \(E_8\) by means of techniques arising from supergravity theory. To any supersymmetric supergravity background there is an associated Lie superalgebra (Killing), obtained from a special class of spinor fields on a Lorentzian spin manifold. In this work, the Lorentzian structure is replaced by a Riemannian manifold, and that of Killing spinor by a nontrivial section of the spinor bundle subjected to a special constraint relating it to Clifford structures. This geometric construction is applied to the spheres of dimensions 7, 8 and 15 to obtain the compact real forms of \(\text{so}(9),\) \(F_4\) and \(E_8\). An interesting fact of this procedure is the relation of the used spheres with the exceptional Hopf fibration.