beiläufig
in passing
Happening or said incidentally and without special emphasis or intention.
Er erwähnte nur beiläufig, dass er am Wochenende umziehen würde.
He mentioned only in passing that he would be moving at the weekend.

casual / in passing
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in passing
Happening or said incidentally and without special emphasis or intention.
Er erwähnte nur beiläufig, dass er am Wochenende umziehen würde.
He mentioned only in passing that he would be moving at the weekend.

The word derives from 'bei' (by/alongside) and 'laufen' (to run/walk). The literal idea is that something happens 'im Vorbeigehen' (in passing), so not as the main thing, but incidentally.
Sie machte eine beiläufige Bemerkung über das Wetter.
She made a casual remark about the weather.
'Beiläufig' (casually/in passing) is very often used as an adverb to describe how an action, usually speaking, is carried out: without emphasis and seemingly unimportant.
Er fragte ganz beiläufig, ob sie am Abend schon etwas vorhatte.
He asked quite casually whether she already had plans for the evening.
The word is often combined with verbs of saying. Typical phrases are 'beiläufig erwähnen' (to mention in passing), 'beiläufig bemerken' (to remark casually), or 'beiläufig fragen' (to ask casually).
Im Gespräch bemerkte sie beiläufig, dass sie eine neue Stelle gefunden hat.
During the conversation, she mentioned in passing that she had found a new job.
random / by chance
'zufällig' and 'beiläufig' can both mean that something is not the main focus or does not seem strongly planned. However, 'zufällig' emphasizes chance: something happens without a plan. 'beiläufig' emphasizes that something is done or said in passing: someone says or does something without special emphasis, often quite calmly in conversation. Learners may confuse the words because both can make something seem not deliberately important. If it is about chance, 'zufällig' fits. If something is only mentioned or done in passing, 'beiläufig' is more natural.

Im Gespräch erwähnte Frau Becker beiläufig, dass sie nächste Woche umzieht.
In conversation, Ms. Becker mentioned in passing that she is moving next week.
Emil findet heute zufällig zehn Euro in seiner alten Jacke.
Today Emil happens to find ten euros in his old jacket.