  An algebra A not encountered in either the usual algebraic varieties or
supervarieties is introduced. A is a graded and deformed version of the
quaternions, with structure similar to that of a Jordan-Lie superalgebra as
defined by Okubo and Kamiya, but it is shown to be neither that of a purely
associative Lie superalgebra, nor that of a purely antiassociative Jordan-Lie
superalgebra. Rather, it exhibits a novel kind of associativity, here called
`ordered graded associativity', that is somewhat `in between' pure
associativity and pure antiassociativity. In addition to graded associativity,
the generators of A obey graded commutation relations encountered in both the
usual Lie superalgebras and in graded Jordan-Lie algebras. They also satisfy
new graded Jacobi identities that combine characteristics of the Jacobis obeyed
by the generators of ungraded Lie, graded Lie and graded Jordan-Lie algebras.
Mainly due to these three features, A is called a `mixed' Jordan-Lie
superalgebra. The present paper defines A and compares it with the Jordan-Lie
superalgebra defined by Okubo and Kamiya.
