\begin{definition}[Definition:Lie Algebra]
Let $\struct {A_R, \oplus}$ be an algebra over a ring.
Then $\struct {A_R, \oplus}$ is a Lie algebra {{iff}}:
:$\forall a \in A_R: a^2 = 0$
:$\forall a, b, c \in A_R: a \paren {b c} + b \paren {c a} + c \paren {a b} = 0$
{{NamedforDef|Marius Sophus Lie|cat = Lie}}
\end{definition}