When to substitute ‘z’ for ‘s’ in American English [duplicate]

I have trouble understanding why some words change "s"-es to "z"-s from BE to AE and some not. For example:

• analyse -> analyze
• characterise -> characterize
• hypnotise -> hypnotize

But:

• compromise -> compromise

Is there any rule to this?

Slightly related: Why isn't "citizen" spelled as "citisen" in British English?

Etymonline contains useful information.

-ize

suffix forming verbs, M.E. -isen, from O.Fr. -iser, from L.L. -izare, from Gk. -izein. English picked up the French form, but partially reverted to the correct Greek -z- spelling from late 16c. In Britain, despite the opposition (at least formerly) of OED, Encyclopaedia Britannica, the “Times of London,” and Fowler, -ise remains dominant. Fowler thinks this is to avoid the difficulty of remembering the short list of common words not from Greek which must be spelled with an -s- (e.g. advertise, devise, surprise).

That last list includes compromise too, as that does not have a Greek root.

The one I’ve had most contact with is baptise/baptize which comes directly from the Greek baptizein and presumably should be spelled with z.

As someone with an interest in letterforms, I've always found z an anomalous letter (the thick stroke goes in the wrong direction) and I far prefer the appearance of these words spelled with an s. That may also be a contributory factor (as well as Fowler’s “difficulty”), even if only subliminally.

The OED^{[paywalled link]} provides this etymological note on the ‑ize suffix, and on how they use that suffix in that dictionary:

Etymology: Cognate with French ‑ise‑r, Italian ‑izare, Spanish ‑izar, < late Latin ‑izāre, ‑īzāre, < Greek ‑ίζειν, formative of verbs.

The Greek verbs were partly intransitive, as βαρβαρίζειν to play the barbarian, act or speak as a barbarian, side with the barbarians, τυραννίζειν to side with the tyrants, partly transitive as καθαρίζειν to purify, clean, θήσαυρίζειν to treasure up. Those formed on national, sectarian, or personal names were primarily intransitive, as Ἀττικίζειν to Atticize in manners, to speak Attic, Φιλιππίζειν to act or speak for Philip, to philippize, Ἑλληνίζειν to ‘do’ the Greek, act as a Greek, speak Greek, Hellenize; also, to make Greek. A few words of this form connected with or used in early Christianity, were latinized already in the 3rd or 4th cent. by Christian writers: such were βαπτίζειν baptizāre, εὐαγγελίζειν euangelizāre, κατηχίζειν catechizāre, σκανδαλίζειν scandalizāre, ἀναθηματίζειν anathēmatizāre, χριστιανίζειν christiānizāre, ἰουδαίζειν iūdaizāre. Others continued to be formed both in ecclesiastical and philosophical use, e.g. canōnizāre, daemonizāre, syllogizāre (Boethius Aristot. Anal.); and this became established as the normal form for the latinizing of Greek verbs, or the formation of verbs upon Greek analogies. In medieval Latin and the modern languages these have been formed also on Latin or modern national names, and the use has been extended to the formation of verbs from Latin adjectives or nouns. This practice probably began first in French; in modern French the suffix has become ‑iser, alike in words from Greek, as baptiser, évangéliser, organiser, and those formed after them from Latin, as civiliser, cicatriser, humaniser. Hence, some have used the spelling ‑ise in English, as in French, for all these words, and some prefer ‑ise in words formed in French or English from Latin elements, retaining ‑ize for those formed < Greek elements. But the suffix itself, whatever the element to which it is added, is in its origin the Greek ‑ιζειν, Latin ‑izāre; and, as the pronunciation is also with z, there is no reason why in English the special French spelling should be followed, in opposition to that which is at once etymological and phonetic. In this Dictionary the termination is uniformly written ‑ize. (In the Greek ‑ιζ‑, the i was short, so originally in Latin, but the double consonant z (= dz, ts) made the syllable long; when the z became a simple consonant, /‑idz/ became īz, whence English /‑aɪz/.)

Adding to the great answer from @AndrewLeach, the answer quoted EtymOnline with

Fowler thinks this is to avoid the difficulty of remembering the short list of common words not from Greek which must be spelled with an -s- (e.g. advertise, devise, surprise).

So, I went in search for the short list and found the following from Lexico, which also adds a bit more to the mix when reading the whole page

In both American and British English, there is a small set of verbs that must always be spelled with -ise at the end. The main reason for this is that, in these words, -ise is part of a longer word element rather than being a separate ending in its own right. For example: -cise (meaning 'cutting) in the word excise; -prise (meaning 'taking') as in surprise; or -mise (meaning 'sending') in promise.

The list they provided is as follows:

• advertise
• compromise
• exercise
• revise
• advise
• despise
• improvise
• supervise
• apprise
• devise
• incise
• surmise
• chastise
• disguise
• prise (meaning ‘open’)
• surprise
• comprise
• excise
• promise
• televise

There are also a few verbs which always end in -yse in British English.

• analyse
• catalyse
• electrolyse
• paralyse
• breathalyse
• dialyse
• hydrolyse
• psychoanalyse

In American English, they are all spelled with the ending -yze