Lie ternary system

A vector space $ \mathfrak m $ with a trilinear composition

$$ \mathfrak m \times \mathfrak m \times \mathfrak m \rightarrow \mathfrak m ,\ \ ( X , Y , Z ) \rightarrow [ X , Y , Z ] , $$

satisfying the following conditions:

$$ [ X , X , Y ] = 0 , $$

$$ [ X , Y , Z ] + [ Y , Z , X ] + [ Z , X , Y ] = 0 , $$

$$ [ X , Y , [ Z , U , V ] ] = $$

$$ = \ [ [ X , Y , Z ] , U , V ] + [ Z , [ X , Y , U ] , V ] + [ Z , U , [ X , Y , V ] ] . $$

If $ \mathfrak g $ is a Lie algebra and $ \mathfrak m \subset \mathfrak g $ is a subspace such that $ [ [ X , Y ] , Z ] \in \mathfrak m $ for any $ X , Y , Z \in \mathfrak m $, then the operation

$$ [ X , Y , Z ] = [ [ X , Y ] , Z ] $$

converts $ \mathfrak m $ into a Lie ternary system. Conversely, every Lie ternary system can be obtained in this way from some Lie algebra.

The category of finite-dimensional Lie ternary systems over the field $ \mathbf R $ is equivalent to the category of simply-connected symmetric homogeneous spaces (see Symmetric space).

References

[1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2]	O. Loos, "Symmetric spaces" , 1 , Benjamin (1969)