  We revisit an order-six linear differential operator having a solution which
is a diagonal of a rational function of three variables. Its exterior square
has a rational solution, indicating that it has a selected differential Galois
group, and is actually homomorphic to its adjoint. We obtain the two
corresponding intertwiners giving this homomorphism to the adjoint. We show
that these intertwiners are also homomorphic to their adjoint and have a simple
decomposition, already underlined in a previous paper, in terms of order-two
self-adjoint operators. From these results, we deduce a new form of
decomposition of operators for this selected order-six linear differential
operator in terms of three order-two self-adjoint operators. We then generalize
the previous decomposition to decompositions in terms of an arbitrary number of
self-adjoint operators of the same parity order. This yields an infinite family
of linear differential operators homomorphic to their adjoint, and, thus, with
a selected differential Galois group. We show that the equivalence of such
operators is compatible with these canonical decompositions. The rational
solutions of the symmetric, or exterior, squares of these selected operators
are, noticeably, seen to depend only on the rightmost self-adjoint operator in
the decomposition. These results, and tools, are applied on operators of large
orders. For instance, it is seen that a large set of (quite massive) operators,
associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained
recently by P. Lairez, correspond to a particular form of the decomposition
detailed in this paper.
