A Moufang set is essentially a doubly transitive permutation group
such that each point stabilizer contains a normal subgroup which is
regular on the remaining vertices; these regular normal subgroups
are called the root groups, and they are assumed to be conjugate
and to generate the whole group. It has been known for some time
that every Jordan division algebra gives rise to a Moufang set with
abelian root groups. The authors extend this result by showing that
every structurable division algebra gives rise to a Moufang set,
and conversely, they show that every Moufang set arising from a
simple linear algebraic group of relative rank one over an
arbitrary field $k$ of characteristic different from $2$ and $3$
arises from a structurable division algebra. The authors also
obtain explicit formulas for the root groups, the $\tau$-map and
the Hua maps of these Moufang sets. This is particularly useful for
the Moufang sets arising from exceptional linear algebraic groups.
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