Then $[\mathfrak g,\mathfrak g]$ is an ideal of $\mathfrak g$, and the [quotient Lie algebra](/theorems/8146) $\mathfrak g/[\mathfrak g,\mathfrak g]$ is abelian.
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Moreover, for every abelian Lie algebra $\mathfrak a$ over $F$ and every Lie algebra homomorphism $\phi:\mathfrak g\to\mathfrak a$, there exists a unique Lie algebra homomorphism
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\begin{align*}
\bar{\phi}: \mathfrak g/[\mathfrak g,\mathfrak g]\to \mathfrak a
\end{align*}
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such that
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\begin{align*}
\phi=\bar{\phi}\circ q,
\end{align*}
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where $q:\mathfrak g\to \mathfrak g/[\mathfrak g,\mathfrak g]$ is the quotient map.