Abstract
We extend the notions of commutativity, ideals, anisotropy, and complemented subtriples of Jordan triple systems to those of Jordan quadruple systems. We show that if [Formula: see text] is a complemented subsystem of an anisotropic commutative Jordan quadruple system [Formula: see text], then [Formula: see text] and its annihilator [Formula: see text] are orthogonal ideals and [Formula: see text]. We also prove that the range of a structural projection on an anisotropic commutative Jordan quadruple system is a complemented ideal and, conversely, a complemented subsystem of an anisotropic commutative Jordan quadruple system is the range of a unique structural projection.
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