In what follows, the following notation will be employed:

• If H and K are subgroups of a group G, the commutator of H and K, denoted by [H, K], is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed.
• If x and y are elements of a group G, the conjugate of x by y will be denoted by ${\displaystyle x^{y}}$ .
• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by C_G(H).

Let X, Y and Z be subgroups of a group G, and assume

${\displaystyle [X,Y,Z]=1}$  and ${\displaystyle [Y,Z,X]=1.}$ 

Then ${\displaystyle [Z,X,Y]=1}$ .^[1]

More generally, for a normal subgroup ${\displaystyle N}$  of ${\displaystyle G}$ , if ${\displaystyle [X,Y,Z]\subseteq N}$  and ${\displaystyle [Y,Z,X]\subseteq N}$ , then ${\displaystyle [Z,X,Y]\subseteq N}$ .^[2]

Hall–Witt identity

If ${\displaystyle x,y,z\in G}$ , then

${\displaystyle [x,y^{-1},z]^{y}\cdot [y,z^{-1},x]^{z}\cdot [z,x^{-1},y]^{x}=1.}$ 

Proof of the three subgroups lemma

Let ${\displaystyle x\in X}$ , ${\displaystyle y\in Y}$ , and ${\displaystyle z\in Z}$ . Then ${\displaystyle [x,y^{-1},z]=1=[y,z^{-1},x]}$ , and by the Hall–Witt identity above, it follows that ${\displaystyle [z,x^{-1},y]^{x}=1}$  and so ${\displaystyle [z,x^{-1},y]=1}$ . Therefore, ${\displaystyle [z,x^{-1}]\in \mathbf {C} _{G}(Y)}$  for all ${\displaystyle z\in Z}$  and ${\displaystyle x\in X}$ . Since these elements generate ${\displaystyle [Z,X]}$ , we conclude that ${\displaystyle [Z,X]\subseteq \mathbf {C} _{G}(Y)}$  and hence ${\displaystyle [Z,X,Y]=1}$ .

• Commutator
• Lower central series
• Grün's lemma
• Jacobi identity

• I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.