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Tits (1962, 1966) made an important observation that relates Lie and Jordan structures. Let L be a Lie superalgebra whose even part contains an -triple {e, f, h}, that is,

DEFINITION 1.6.– An -triple e, f, h is said to be “good” if ad(h) : L → L is diagonalizable and the eigenvalues are only –2, 0, 2.

In such a case, L = L_{– 2} + L₀ + L₂ decomposes as a direct sum of eigenspaces. We can define a new product in L₂ by:

With this new product, J = (L₂, ○) becomes a Jordan superalgebra.

Moreover, (Tits 1962, 1966; Kantor 1972) and (Koecher 1967) showed that every Jordan superalgebra can be obtained in this way. The corresponding Lie superalgebra is not unique, but any two such Lie superalgebras are centrally isogenous, that is, they have the same central cover. Let us recall the construction of L = TKK(J), the universal Lie superalgebra in this class (see Martin and Piard (1992)).

CONSTRUCTION.– Consider J a unital Jordan superalgebra, and {ei}_i∈I a basis of J.

Let

Define a Lie superalgebra K by generators and relations

This Lie superalgebra has a short grading K = K_–1 + K₀+ K₁ where

K is the universal Tits–Kantor–Koecher Lie superalgebra of the unital Jordan superalgebra J: