Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so-called mixed groups of type F 4 in characteristic 2, obtained as the fixed point set under a suitable involution.

For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E 6 . We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set.It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F 4 , (2) Moufang sets of type 2 E 6 , (3) hermitian Moufang sets of type C 4 , and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2 E 6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type 2 E 29 6,1 ; the explicit description of those Moufang sets was not yet known.

Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so-called mixed groups of type F 4 in characteristic 2, obtained as the fixed point set under a suitable involution. Setup Chevalley groupsWe briefly recall some basics about Chevalley groups that we will need in the sequel. Our main reference is [6].

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