Since F_4 is reductive, the quotient SO(26)/F_4 is an affine algebraic variety. Is there any explicit description of this variety, that would also include the action of SO(26) on? Say, as a closed subset of an affine space? Just to give an idea what I mean by this, let us consider the quotient GL(n)/O(n). It is easy to see that GL(n)/O(n) is isomorphic to the subset S\subseteq GL(n), consisting of all symmetric matrices. The isomorphism is induced by the map g\mapsto gg^t, hence the action of GL(n) on S is defined as s^g=gsg^t, s\in S, g\in GL(n).
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