The class of Banach lattices is not primary

Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$-spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space L such that C(L) X ⊕ X̃ and neither X nor X̃ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$-spaces is not primary.