  In this paper we revisit and extend the constructions of Glauberman and Doro
on groups with triality and Moufang loops to Hopf algebras. We prove that the
universal enveloping algebra of any Lie algebra with triality is a Hopf algebra
with triality. This allows us to give a new construction of the universal
enveloping algebras of Malcev algebras. Our work relies on the approach of
Grishkov and Zavarnitsine to groups with triality.
