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How does SageMath compute the Jordan form of a block diagonal matrix?

Let M be a block diagonal matrix diag(B1,,Bm). Let (pi) be the orthogonal projections such that Bi=piMpi. If you compute the Jordan form of M using SageMath as follows:

sage: jf, q = M.jordan_form(transformation=True)

Question: Is it true that q commutes with pi for all i?

It would be true if SageMath computes the Jordan form of M by computing the Jordan form of each Bi.