1-complemented subspaces of spaces with 1-unconditional bases

We prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y subset of or equal to X has no bands isometric to l(2)(2) and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X. We completely characterize 1-complemented subspaces and norm-one projections in complex spaces l(p)(l(q)) for 1 less than or equal to p, q < infinity. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to l(p) for some 1 less than or equal to p < infinity then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.