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## Homework Statement

Let [itex]\mathfrak{g}[/itex] be any lie algebra and [itex]\mathfrak{h}[/itex] be any ideal of [itex]\mathfrak{g}[/itex].

The canonical homomorphism [itex]\pi : \mathfrak{g} \to \mathfrak{g/h}[/itex] is defined [itex]\pi (x) = x + \mathfrak{h}[/itex] for all [itex]x\in\mathfrak{g}[/itex].

For any ideal [itex]\mathfrak{f}[/itex] of the quotient lie algebra [itex]\mathfrak{g/h}[/itex], consider the inverse image of [itex]\mathfrak{f}[/itex] in [itex]\mathfrak{g}[/itex] relative to [itex]\pi[/itex], that is: [tex]\pi ^{-1} (\mathfrak{f}) = \{X\in\mathfrak{g} : \pi (X)\in \mathfrak{f} \} .[/tex]

Prove that [itex]\pi ^{-1} (\mathfrak{f})[/itex] is an ideal of the lie algebra [itex]\mathfrak{g}[/itex].

## The Attempt at a Solution

See below

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