On embeddings in the intersection $$X\cap L_{\infty }$$

Let X be a separable rearrangement invariant space on \((0,\infty )\). If the intersection \((X \cap L_{\infty })(0,\infty )\) contains a complemented subspace isomorphic to \({\ell }_2\), then X contains a complemented sublattice lattice-isomorphic to \({\ell }_2\). Moreover, we prove that the space \((X+L_{\infty })(0,\infty )\) cannot be isomorphically embedded into \((X \cap L_{\infty })(0,\infty )\) as a complemented subspace provided that X has nontrivial Rademacher cotype.