The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the case of an Engel group. The Engel identity is the defining condition of an Engel group.

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A Lie ring ${\displaystyle L}$ is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket ${\displaystyle [x,y]}$, defined for all elements ${\displaystyle x,y}$ in the ring ${\displaystyle L}$. The Lie ring ${\displaystyle L}$ is defined to be an n-Engel Lie ring if and only if

• for all ${\displaystyle x,y}$ in ${\displaystyle L}$, the n-Engel identity

${\displaystyle [x,[x,\ldots ,[x,[x,y]]\ldots ]]=0}$ (n copies of ${\displaystyle x}$), is satisfied.^[1]

In the case of a group ${\displaystyle G}$, in the preceding definition, use the definition [x,y] = x⁻¹ • y⁻¹ • x • y and replace ${\displaystyle 0}$ by ${\displaystyle 1}$, where ${\displaystyle 1}$ is the identity element of the group ${\displaystyle G}$.^[2]

• Adjoint representation
• Efim Zelmanov
• Engel's theorem