Let $F$ be a field, let $\mathfrak g$ be a [Lie algebra](/page/Lie%20Algebra) over $F$ with bracket $[\cdot,\cdot]_{\mathfrak g}$, and let $\mathfrak i \subset \mathfrak g$ be an ideal. Define a map
\begin{align*}
[x+\mathfrak i,y+\mathfrak i]_{\mathfrak g/\mathfrak i}=[x,y]_{\mathfrak g}+\mathfrak i
\end{align*}
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for all $x,y\in\mathfrak g$. Then this map is well-defined and makes the quotient [vector space](/page/Vector%20Space) $\mathfrak g/\mathfrak i$ into a Lie algebra over $F$.