The space of all $n$-dimensional complex Lie algebras $\mathfrak g\simeq\mathbb C^n$ can be identified with “the Lie algebra variety” $\mathcal L_n$, which is the closed subset of $V_n := \mathfrak g^*\wedge \mathfrak g^*\otimes \mathfrak g$ cut out by $n^4$ quadratic equations in the structure constants $\mu^k_{ij} := \langle \mu(e_i,e_j), e_k\rangle$ \begin{equation} \mu^s_{ir}\mu^r_{jk}+\mu^s_{jr}\mu^r_{ki} + \mu^s_{kr}\mu^r_{ij} = 0 \tag{1}%{Jacobi in the structure constants} \end{equation} imposed by the Jacobi identity. According to Jorge Lauret. On the moment map for the variety of lie algebras, the isomorphism class of a Lie algebra $\mu\in \mathcal L_n$ is the orbit $\operatorname{GL}(n)\cdot\mu$ under the ‘change of basis’ action of $\operatorname{GL}(n)$ on $V_n$.
I am confused here because as far as I know, by definition, two Lie algebras $\mathfrak g_{\mu}$ and $\mathfrak g_{\mu’}$ are isomorphic only if there is an invertible linear transformation $h:\mathfrak g_\mu \rightarrow \mathfrak g_{\mu’}$ such that the brackets $\mu$ and $\mu’$ are related by
$$h(\mu(e_i,e_j)) = \mu’(h(e_i),h(e_j))\tag{2},$$
i.e. if $h \in \operatorname{Aut}(\mathfrak g_\mu) \subset \operatorname{GL}(\mathfrak g_\mu)$. Thus, the isomorphism class of $\mu\in\mathcal L_n$ should be given by the orbit $\operatorname{Aut}(\mathfrak g)\cdot \mu$ and not by $\operatorname{GL}(n)\cdot \mu$. Did Lauret mean a different notion of isomorphism of Lie algebras, such as only requiring the existence of the invertible linear transformation $h$ and not $(2)$?
Furthermore, I rewrote $(1)$ under the action $\mu\mapsto h\cdot \mu$, where $h\in \operatorname{GL}(n)$ and $h\cdot \mu(x,y) = h\mu(h^{-1}x,h^{-1}y)$, and obtained \begin{equation} (h\cdot\mu^s_{ir})(h\cdot\mu^r_{jk})+(h\cdot\mu^s_{jr})(h\cdot\mu^r_{ki}) + (h\cdot\mu^s_{kr})(h\cdot\mu^r_{ij}) = 0, \tag{3} \end{equation} where $h\cdot\mu^s_{ir}=\langle h\mu(h^{-1}e_i,h^{-1}e_j), e_k\rangle$. However, it’s not clear to me $(3)$ is satisfied if $(1)$ is, i.e. that $\mathcal L_n$ is indeed preserved by the action of $\operatorname{GL}(n)$.