$$\mathcal{V}$$ a grading Definition 2.2
$$V_\alpha$$ a layer of a grading Definition 2.2
$$\Omega$$ the set of weights of a grading Definition 2.2
$$f_* \mathcal V$$ a push-forward grading by a homomorphism $$f$$ Definition 2.4
$$f_* \Phi(\mathcal V)$$ a push-forward grading including an automorphism $$\Phi$$ Definition 2.4
$$\Aut(\mathfrak{g})$$ the automorphism group of $$\mathfrak{g}$$ Definition 2.4
$$\langle X\rangle$$ the span of $$X$$ Paragraph
$$\der(\mathfrak{g})$$ the derivation algebra of $$\mathfrak{g}$$ Paragraph