A Peirce inner ideal J in an anisotropic Jordan * -triple A gives rise to a Peirce grading (J0, J1, J2) of A by defining J0 = J ⊥ , J1 = Ker(J ) ∩ Ker(J ⊥ ), J2 = J, where J ⊥ is the set of elements a of A for which {J a A} is equal to {0} and Ker(J ) is the set of elements a of A for which {J a J} is equal to {0}. It is shown that conversely, when A is a JBW * -triple factor, for each Peirce grading (J0, J1, J2) of A such that both J0 and J2 are non-zero, both J0 and J2 are Peirce inner ideals the corresponding Peirce decompositions of A being given by (J0)0 = J2, (J0)1 = J1, (J0)2 = J0; (J2)0 = J0, (J2)1 = J1, (J2)2 = J2. Mathematics Subject Classification (2000). Primary 17C65; Secondary 46L70.

Let (B + , B − ) be an involutive grading of a JBW * -triple factor A with associated involutive triple automorphism φ. When the JBW * -subtriple B + of A is not a JBW * -triple factor there exists a non-zero Peirce weak * -closed inner ideal J in A with Peirce spaces J 0 , J 1 , and J 2 such thatWhen both B + and B − are JBW * -triple factors it is shown that either the situation reduces to that above with J 0 or J 2 equal to zero or, in the case that B + (or, by symmetry, B − ) contains a unitary tripotent v, that v is unitary in A, andwhere H (A 2 (v), φ) is the JBW * -algebra of φ-invariant elements in the JBW * -algebra A 2 (v), and S(A 2 (v), φ) is the JBW * -triple of −φ-invariant elements of A 2 (v). In the special case in which A is a discrete W * -factor it is shown that such a unitary tripotent always exists in B + (or B − ), thereby completing the description of involutive gradings in this case.