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We develop new machinery for producing decomposability tests for involutivesolutions to the Yang-Baxter equation. It is based on the seminaldecomposability theorem of Rump, and on "cabling" operations on solutions andtheir effect on the diagonal map. Our machinery yields an elementary proof of arecent decomposability theorem of Camp-More and Sastriques, as well as originaldecomposability results. It also provides a conceptual interpretation (usingthe braces language) of the Dehornoy class, a combinatorial invariant naturallyappearing in the Garside-theoretic approach to involutive solutions.

Involutive Yang-Baxter: cabling, decomposability, Dehornoy class