The class I[D[I]] is composable but not splittable, answering Karpilovskij's question, and every infinite composable class avoiding an increasing or decreasing permutation is splittable.
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The Relation between Composability and Splittability of Permutation Classes
The class I[D[I]] is composable but not splittable, answering Karpilovskij's question, and every infinite composable class avoiding an increasing or decreasing permutation is splittable.