Abstract
We present systems of parabosons and parafermions in the context of Lie algebras, Lie superalgebras, Z₂× Z₂-graded Lie algebras and Z₂× Z₂-graded Lie superalgebras. For certain relevant Z₂× Z₂-graded Lie algebras and Z₂× Z₂-graded Lie superalgebras, some structure theory in terms of roots and root vectors is developed. The short root vectors of these algebras are identified with parastatistics operators. For the Z₂× Z₂-graded Lie algebra so_q(2n+1), a system consisting of two ensembles of parafermions satisfying relative paraboson relations are introduced. For the Z₂× Z₂-graded Lie superalgebra osp(1,0|2n₁,2n₂), a system consisting of two ensembles of parabosons satisfying relative parafermion relations are introduced.