ИСТИНА

It is shown how, using the Clifford algebra, it is possible to turn an ordinary commutator into an anticommutator and vice versa. This is illustrated on ordinary bosons and fermions. Next, we consider Lie superalgebras, which are superextensions of the complex Lie algebra so(5,C) and its real forms: de Sitter so(1,4) and anti de Sitter so(2,3). There are two types of such superextensions: with \mathbb{Z}_2- and \mathbb{Z}_2\times\mathbb{Z}_2- gradations. The Lie \mathbb{Z}_2-superalgebras are the usual Lie seperalgebras, and the Lie \mathbb{Z}_2\times\mathbb{Z}_2-superalgebras were presented by the speaker in 2014. It is shown how these superalgebras with different gradations are related to each other using the Clifford algebra. For the first time, the correct concept of compact real forms is introduced for both \mathbb{Z}_2- and \mathbb{Z}_2\times\mathbb{Z}_2-graded superextensions of so(5;C). For this purpose, we use a cliffonic dressing with a Hermitian imaginary unit (the usual imaginary unit is an anti-Hermitian one). We consider also a superextension in the case of the superalgebra osp(1|2n;C)