While browsing the arxiv, I came upon a cool new paper on the group F4. The paper contains a plethora of mathematical treasures, including: the exceptional Jordan algebra, OP^2 and the generalized F4 Euler angle construction I mentioned to Kea and Carl earlier this year. Pierre Ramond explored these issues in the context of M-theory back in 1998 (hep-th/9808190). More recently, the physical context has been extended to extremal black holes in N=2 Maxwell-Einstein supergravities (hep-th/0512296).
In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.