Yes, the definition of canonical sequences is something like that I want. However, as Rokicki pointed out, it does not cover cases like U2 D2 F2 B2 = F2 B2 U2 D2, which I feel still is a trivial identity, because with basic transformations I can get:
U2 D2 F2 B2 = E2 y2 S2 z2 = y2 E2 S2 z2 = y2 S2 E2 z2 = (...) = S2 z2 E2 y2 = F2 B2 U2 D2.
Canonical sequences are based on a very simple principle: adjacent commuting moves have a single
preferred order.
For the half-turn and quarter-turn metric, the close correspondence between the count of sequences
of length $n$ and positions at distance $n$ with this simple definition means that search can be
performed effectively using this simple definition.
In the slice-turn metric, things get a little messier. S2 and E2 commute in the slice-turn metric, but
the reduction to a single canonical sequence is more complicated since they cross syllables. In the
HTM and QTM, the only commuting moves are in the same syllable so the reduction is straightforward.
However, in the STM, we end up with things like this:
U2 S2 E2 F2
If "S2 E2" is the preferred order for those commuting moves, we cannot reduce this further by
inspection only of adjacent moves. But if we can flip those, we get
U2 E2 S2 F2
which of course can be reduced to
D2 B2 z2
but this reduction requires a bit more complexity. And as it turns out, the reduction in search
space, even in the slice-turn metric, is not all that great. Is it worth it? Perhaps.
So I guess what we lack most here is how you define "triviality" of an identity. The pair
pointed out by Lucas comes from the fact that the position he shows is antisymmetric, so
the second move sequence is just the inverse of the first but reoriented.
