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