I=image(f,S)
If $S$ is of type FGLieSubAlgebra, then image(f,S) is of type FGLieSubAlgebra. If $S$ is an instance of LieSubAlgebra, but not of FGLieSubAlgebra, then image(f,S) is of type LieSubAlgebra. Otherwise, image(f,S) is of type LieSubSpace.
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The source of this document is in GradedLieAlgebras/doc.m2:1195:0.