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I think Betza suggested also other uses for the brackets, like z[F,W] for a slider that alternats W and F steps in a crooked way, but this distinction could also have been made by using other separators than comma for that, e.g. [W/F].

He did indeed. The alternation modifier was in fact a, contrasting q which alternated circularly if followed by a set of brackets; t is also defined there, as is g (for ‘go’ — equivalent to the proposed [X-Y] to t[]'s [X~Y]) which covers the mao case (though conflicts with the Grasshopper usage).

writing the Griffon as F&fR or [F-fR] assumes the move can also be terminated without making all its legs, after just the F step.

There is technically another interpretation which would not conflict with the mao (and would obviate the need for Betzan g[] in the common case — though the original rhino (mao+wazir) would still need either the distinction or expliit compounding), which you've mentioned before: consider slider legs to move 0 or more rather than 1 or more, while leapers are still exactly 1. The arguably more complex piece that follows a gryphon's path but must move at least two spaces then gets a suitably more complex notation (e.g. Betza‐style t[FWR] or the like). This would also allow e.g. Tim Stiles' doubly‐bent Fox to be trivially t[WBW]. Of course with still more complex paths (t[WFR]?) the same considerations apply, though counting to 3 or more starts to be complicated for humans too so more specific notations of the likes of what are being discussed here are probably in order anyway.

What if doubling a direction made it absolute instead of relative?

As HG points out, duplication is already in use for other things; but in principle one could add a punctuation mark (maybe an apostrophe or an exclemation mark) to mark a direction as absolute rather than relative, which would be roughly equivalent

considering a certain grouped sequence of directional modifiers plus atoms as a 'crooked atom'

This is the interpretation I've been coming to for most chess‐variant pieces in general. Some kind of (for me, radial‐step — Nightriders have more in common with Dabbabariders than with Rooks imo) path and, independently, a set of constraints on that path, be it leaping, limited range, skipping squares, hopping, etc. And modality (movement, capture, or other special effects such as relaying or rifle‐capture) as a third factor on top of that. Works for most of the pieces people actually use afaict.

So the Ship would be the 'Narrow Griffon', like vN is the Narrow Knight.

I second this and the v[F-R]‐or‐equivalent notation, if a bracket‐style notation is being adopted, and if it's easy p[F‐R] and the like look nice too.

Worth noting as well that Betza also made a similar extrapolation in defining the a[WF]4 on the above page (just above the Two Sets, Four Boards heading)