Remark 1

Let $e_1= (\mathfrak{g},\mathfrak{h}_1,B_1,(X_{\alpha}^1)_{\alpha\in B_1}),e_2= (\mathfrak{g},\mathfrak{h}_2,B_2,(X_{\alpha}^2)_{\alpha\in B_2})$, $e'_2= (\mathfrak{g},\mathfrak{h}_2,B_2,(Y_{\alpha}^2)_{\alpha\in B_2})$ be framings of $\mathfrak{g}$, and $s$ (resp. $s'$) an element of Aut$_0(\mathfrak{g})$ that transforms $e_1$ to $e_2$ (resp. $e'_2$). Then $s|\mathfrak{h}_1=s'|\mathfrak{h}_1$. Indeed, $s^{'-1}s\in$ Aut$_0(\mathfrak{g},\mathfrak{h}_1)$ and $s^{'-1}s(B_1) = B_1$, so $\varepsilon (s^{'-1}s) = 1$.