Is it obligatory to inform the husband of his wife’s illicit relationship with another man, given that there is a daughter and that the husband is impetuous and loves his wife very much?

You must verify before accusing others. If there is clear evidence of adultery or repeated unlawful seclusion (khalwa), then the steps are:

First, advise them and remind them of Allah. If they do not respond, then:

Second, threaten to expose their matter and inform official authorities or relatives.

Third, if that does not work and you have evidence, then you are obligated to inform someone who can prevent this wrongdoing.

Proving adultery requires four upright male witnesses who saw the act firsthand. This is very difficult, as it is done to protect reputations. Whoever accuses without evidence is a slanderer (qādhif).

If you are certain of adultery but do not have witnesses, you can inform someone who can change the wrongdoing with a message, without exposing yourself to punishment.