  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 E6. 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 F4, (2) Moufang sets of type 2E6, (3) hermitian
Moufang sets of type C4, 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 2E6 that we obtain in case (2) are exactly those
corresponding to linear algebraic groups of type 2E6,1^29; the explicit
description of those Moufang sets was not yet known.

Group Theory