A subspace J of an anisotropic Jordan*-triple A is said to be an inner ideal if Ä 4 the subspace J A J is contained in J. An inner ideal J in A is said to be Ž . complemented if A is equal to the sum of J and the kernel Ker J of J, defined to Ä 4 be the subspace of A consisting of elements a in A for which J a J is equal to Ä 4 H 0 . The annihilator J of an inner ideal J in A is the inner ideal consisting of Ä 4 Ä 4
H elements a in A such that J a A is equal to 0 . When both J and J are Ž .
Ž H . complemented, A can be decomposed into the direct sum of J, Ker J l Ker J and J H . Modulo six of the generalized Peirce relations this decomposition is a grading of A of Peirce type. Since an inner ideal in a JBW U -triple is complemented if and only if it is weak U -closed, the result described above applies to all weak U -closed inner ideals J in a JBW U -triple A. Furthermore, it can be shown that in this case all except five of the generalized Peirce relations hold, and an example is given of a weak U -closed inner ideal in a JBW U -triple for which all five fail to hold, thereby showing that the result is the best possible. It is also shown that the condition that a weak U -closed inner ideal in a JBW U -triple A leads to a grading of A which is of Peirce type is equivalent to several other conditions, all of a topological, rather than algebraic, nature. These results are applied to W U -algebras, spin triples, and the bi-Cayley triple.