vier
four
The cardinal number 4; the quantity between three and five.
Das Auto hat vier Räder.
The car has four wheels.

four
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four
The cardinal number 4; the quantity between three and five.
Das Auto hat vier Räder.
The car has four wheels.

The number 'vier' (four) is the basis for many other useful words. These include 'das Viertel' (a district or 1/4), 'viertens' (fourthly, as a list item), and 'viereckig' (an adjective for a shape with four corners).
Schneide den Apfel bitte in vier Viertel.
Please cut the apple into four quarters.
The idiom 'unter vier Augen' means that something is discussed confidentially between only two people. The number 'vier' (four) refers to the eyes of the two people present.
Ich möchte das Thema nach der Besprechung mit dir unter vier Augen diskutieren.
I would like to discuss the topic with you in private after the meeting.
The expression 'auf allen vieren' describes movement on hands and knees. In this expression, the form 'vieren' is the dative plural, which is a fixed part of the idiom here.
Er rutschte auf allen vieren unter dem Tisch, um sein Handy zu suchen.
He slid on all fours under the table to look for his phone.
private
'unter vier Augen' and 'privat' can both mean that something is not intended for other people. However, 'unter vier Augen' describes a specific conversation situation: only two people are speaking with each other, without any listeners. 'privat' is more general and means that something belongs to personal life or is not public. A conversation can be private, but it is only 'unter vier Augen' if exactly two people are involved.

Der Chef möchte das Problem unter vier Augen mit mir besprechen.
The boss wants to discuss the problem with me in private.
Nach der Arbeit beantwortet Frau Müller keine privaten Nachrichten mehr.
After work, Ms. Müller no longer answers private messages.