Necessary and sufficient conditions for the existence of a contractive projection from the predual \(A_*\) of a JBW*-triple onto the closed linear span of a set of elements of \(A_*\) whose support tripotents form a collinear system are provided. To this end, the author studies algebraic conditions for the range of a contractive projection on a JBW*-triple \(A\) to be a subtriple; in particular, these results also provide criteria for the range of a normal contractive projection on \(A\) to be a Hilbert space.