The $g_{a}$-distance to the origin is $|(\eta_{1},\eta_{2},\mu)|_{a}=\sqrt{a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2}}$. A variant

$$\varrho=|(y_{1},y_{2},\mu)|_{a}^{\prime}=(\frac{A}{\mathbb{A}}a_{p\bar{q}}y_{p}y_{q}+A\mu^{2})^{1/2},\quad\mathbb{A}=A+|\text{Im}(a_{1\bar{2}})|^{2}.$$

stands for the distance function for the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{y}$

(4.3)

$$\begin{split}g_{a}^{\prime}=\frac{A}{\mathbb{A}}(a_{1\bar{1}}dy_{1}^{2}+2\text{Re}(a_{1\bar{2}})dy_{1}dy_{2}+a_{2\bar{2}}dy_{2}^{2})+A|d\mu|^{2}.\end{split}$$

Let $S$ is $R=\text{dist}_{g_{a}}(\cdot,S)$. The parameter $R+A^{-1/2}$ is relevant for regularity scales.