  We construct an universal enveloping algebra associated to the ternary
extension of Lie (super)algebras called Lie algebra of order three. A
Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown
that this universal enveloping algebra can be endowed with a structure of Hopf
algebra. The study of the dual of the universal enveloping algebra enables to
define the parameters of the transformation of a Lie algebra of order three. It
turns out that these variables are the variables which generate the
three-exterior algebra.
