We denote $\vec{\mu}=(\mu_{1},\mu_{2},\eta)$ and $|\vec{\mu}|_{a}=\sqrt{a_{ij}\mu_{i}\mu_{j}+A|\eta|^{2}}$ is the $g_{a}$-distance to the origin. A variant $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}=\sqrt{a_{ij}\mu_{i}\mu_{j}+Ay^{2}}$ stands for the distance in the $g_{a}^{\prime}$-metric on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$

(3.3)

$$g_{a}^{\prime}=a_{ij}d\mu_{i}d\mu_{j}+Ady^{2}=a_{ij}d\mu_{i}d\mu_{j}+A|d\text{Im}(\eta)|^{2}.$$

Another useful length parameter is $\ell=\text{dist}_{g_{a}}(\cdot,\mathfrak{D})+A^{-1/4}$ which is relevant for regularity scales.