
beieinander
close together
Spatially close together or next to each other.
Die beiden Schwestern saßen eng beieinander auf dem Sofa.
The two sisters sat close together on the sofa.
together
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close together
Spatially close together or next to each other.
Die beiden Schwestern saßen eng beieinander auf dem Sofa.
The two sisters sat close together on the sofa.

together
Being gathered in a group or in company.
An Weihnachten ist immer die ganze Familie gemütlich beieinander.
At Christmas, the whole family is always together in a cozy way.

not quite with it
Colloquial for being confused, unfocused, or not clear-headed.
Nach nur drei Stunden Schlaf war er am nächsten Morgen nicht ganz beieinander.
After only three hours of sleep, he was not quite with it the next morning.
The word is made up of the preposition 'bei' (near/with) and the pronoun 'einander' (each other). So it literally means 'near each other'.
Die beiden Häuser stehen sehr dicht beieinander.
The two houses stand very close together.
As an adverb, 'beieinander' (together) describes the way people or things are positioned in relation to one another. It usually comes after the conjugated verb.
Nach dem Streit saßen sie lange schweigend beieinander.
After the argument, they sat together in silence for a long time.
When referring to a social group, 'beieinander' (together) is often combined with adverbs that describe the atmosphere, e.g. 'gemütlich beieinander sitzen' (to sit together cozily) or 'friedlich beieinander sein' (to be together peacefully).
Am Lagerfeuer saß die ganze Gruppe noch lange gemütlich beieinander.
The whole group sat together cozily around the campfire for a long time.
The expression 'nicht ganz beieinander sein' (not to be quite with it) is a fixed colloquial phrase. It means that someone is confused or unfocused.
Er redet heute nur wirres Zeug, ich glaube, er ist nicht ganz beieinander.
He's only talking nonsense today; I think he's not quite with it.
together
When people are in the same place, both zusammen and beieinander can work. Beieinander emphasizes physical proximity or being present together, while zusammen can also express joint action or a connection. Use beieinander when the main point is that no one is physically separated from the others.

Nach dem Essen sitzen wir noch beieinander im Wohnzimmer.
After dinner, we are still sitting close together in the living room.
Nach dem Essen sitzen wir noch zusammen im Wohnzimmer.
After dinner, we are still sitting together in the living room.