Contractivity of Projections Commuting with Inner Derivations on JBW-triples

It is shown that if P is a weak*-continuous projection on a JBW*-triple A with predual A * , such that the range PA of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of PA , then P is contractive if and only if it commutes with all inner derivations of PA . This provides characterizations of 1-complemented elements in a large class of subspaces of A * in terms of commutation relations.