Q=L/x
Consider first the case where $L$ has zero differential, and where $L$ is finitely presented as a quotient of a free Lie algebra $F$. In this case, the output $Q$ is also finitely presented as a quotient of $F$.
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In case $L$ has a non-zero differential, the program adds relations depending on the fact that the ideal should be invariant under the differential. These extra (non-normalized) relations may be looked upon using describe(LieAlgebra). Observe that $D$ is not free in this example, see differentialLieAlgebra.
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If the input Lie algebra $L$ is given as a finitely presented Lie algebra $M$ modulo an ideal $J$ that is not (known to be) finitely generated (e.g., the kernel of a homomorphism ), then the output Lie algebra $Q$ is presented as a quotient of a finitely presented Lie algebra $N$ by an ideal $I$, where $N$ is given as $M$ modulo a lifting of the input list $x$ to $M$, and $I$ is the image of the natural map from $M$ to $N$ applied to $J$, see image(LieAlgebraMap,LieSubSpace).
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The source of this document is in GradedLieAlgebras/doc.m2:1724:0.