Learn English – Why do some non-English words become English words

I would say that my answer about when to italicize loaned words and phrases is also the most appropriate answer to this question:

I think this is a case where authors can decide for themselves where to draw the line. Or, if the writing is for a certain publication, the editors will have a policy for whether a given word should or should not [have diacritics].

(However, that answer got zero votes, so who knows? I still think it is a good answer.)

Various publications will have strict rules about what diacritics are used and when. For example, The New Yorker always uses diaeresis. Each publication seems to have its own rules for bolding, headers, section numbering, hyphenation, capitalization, reference citation, and so on. And there is no definitive format.

I believe that diacritics fall into this category, as evidenced by their generally inconsistent use (though internally consistent within many publications).

In short, your only option seems to be exegese (or possibly exegeze, but probably not), if you must have a verb that would be acceptable to the OED.

But it should be noted that, for what is probably the majority of words on -sis, no verb is ever used; perhaps recasting your sentence would be a better idea, e.g. she interpreted the passage. Using the Greek infinitive exegeisthai is a bad idea that will send classicists off screaming.

The only traditional way of forming a verb based on a Greek word on -sis that I am aware of is -se:

Parenthesis => parenthese

Diagnosis => diagnose

Paralysis => paralyse

Basis/-se => base

Metamorphosis => metamorphose

Occasionally, modern forms on -sise are created, as in those based on -thesis (hypothesise, synthesise); I can't think of any other example. Those few Greek words on -sis whose noun-equivalents already end on -se in English also use -se for the verb, as in phrase and base.

For forms on -ise and -yse, you may encounter -ize and -yze in America, respectively, and possibly sometimes elsewhere too, so exegeze may be considered a valid alternative depending on your local tradition. However, this strictly has nothing to do with the formation of an English word based on Greek, but rather on regional variations within English. Lastly, I do not believe Americans normally do this with words on -ese/-eze (they stick with -ese: only after -i- and -y- is this z commonly used), but that's your call.

Your examples ostracize and baptize are different, because they have somehow retained or regained the z that was originally there in Greek (ostrakizein, baptizein), as opposed to in exegeisthai and most other verbs. The same applies to apologize/apologizein. This confusion is the reason (most?) British publishers and style books use -ise/-yse for verbs based on Greek nouns on -sis (Oxford and Cambridge alike, I think), like analyse; but they do often do use -ize for other Greek verbs, like baptize, and for words taken from non-Greek stems, like immunize, realize, colonize (either Oxford or Cambridge—I forgot). They usually do not, however, use -yze where Greek had no -uzein (so practically never). I know, it is a bit tiresome having to remember the exact origin of such words—I am usually too lazy to do so, I must confess.

The -i- in *exegise/exegize is not really defensible, because that is normally not done with words derived from -esis, but only with those derived from -isis. The -ize in energize comes from Greek -(e)izein, as energy comes from energ(e)ia; the -i- in apologize from Greek -izein as in apology, from apologia. The -y in English represents -(e)ia in most Greek (and Latin) words, and hence nouns on -y in English have verbs on -ise/ize.