Witt's identity

This fact is related to: commutator calculus
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Statement

Let $a,b,c$ be elements of an arbitrary group $G$. Then:

$[[a,b^{-1}],c{]}^b\cdot [[b,c^{-1}],a{]}^c\cdot [[c,a^{-1}],b{]}^a=1$

where $[x,y]=x^{-1}{y}^{-1}xy$ and $x^y=y^{-1}xy$

Related results

• Jacobi identity
• Three subgroup lemma