\begin{equation*} \begin{aligned} f(x,y,z) & = [C_kS_i^2S_j^2(1-\varphi _{ij}^2) + C_iS_iS_j^2S_k(\varphi _{ik} + \varphi _{ij}\varphi _{jk}) + C_jS_i^2S_jS_k(\varphi _{jk} + \varphi _{ij}\varphi _{ik})] e^{2r_i+2r_j+r_k} \\ & = a_1 \frac{z+1}{2}\frac{(x-1)^2}{4}\frac{(y-1)^2}{4} +b_2 \frac{x^2-1}{4}\frac{(y-1)^2}{4}\frac{z-1}{2}+b_3 \frac{y^2-1}{4} \frac{(x-1)^2}{4}\frac{z-1}{2} \\ & = \frac{a_1(x-1)^2(y-1)^2(z+1)+ b_2(x^2-1)(y-1)^2(z-1) +b_3(x-1)^2(y^2-1)(z-1)}{32} \\ & = \frac{1}{32}\Bigl[(a_1+b_2+b_3)x^2y^2z+(a_1-b_2-b_3)x^2y^2+(-2a_1-2b_2)x^2yz+(-2a_1+2b_2)x^2y \\ & \quad +(a_1+b_2-b_3)x^2z+(a_1-b_2+b_3)x^2+(-2a_1-2b_3)xy^2z+(-2a_1+2b_3)xy^2+4a_1xyz+4a_1xy \\ & \quad +(-2a_1+2b_3)xz+(-2a_1-2b_3)x+(a_1-b_2+b_3)y^2z+(a_1+b_2-b_3)y^2+(-2a_1+2b_2)yz \\ & \quad +(-2a_1-2b_2)y+(a_1-b_2-b_3)z+(a_1+b_2+b_3)\Bigr]. \end{aligned}\end{equation*}

\begin{equation*} \begin{aligned} g_1(x,y,z) & = [(C_iC_j + \varphi _{ij}S_iS_j)^2 - 1]e^{2r_i+2r_j} \\ & = (\frac{e^{2r_i}+ 1}{2}\frac{e^{2r_j}+ 1}{2} + \varphi _{ij} \frac{e^{2r_i}- 1}{2}\frac{e^{2r_j}- 1}{2} )^2 - e^{2r_i+2r_j} \\ & = (\frac{x+1}{2}\frac{y+1}{2}+ \varphi _{ij}\frac{x-1}{2}\frac{y-1}{2})^2 - xy \\ & = \frac{1}{16}\Big[(1+\varphi _{ij})^2x^2y^2+2(1-\varphi _{ij}^2)(x^2y+xy^2)+(1-\varphi _{ij})^2(x^2+y^2) \\ & \ \ \ \ \ \ \ \ +4(1+\varphi _{ij}^2)xy+2(1-\varphi _{ij}^2)(x+y)+(1+\varphi _{ij})^2\Big]-xy. \end{aligned}\end{equation*}

\begin{equation*} \begin{aligned} g_2(x,y,z)& = e^{2r_i+2r_j+2r_k}\Delta \\ & = e^{2r_i+2r_j+2r_k}[(2+2\varphi _{ij}\varphi _{jk}\varphi _{ik})S_i^2S_j^2S_k^2 + (1-\varphi _{ij}^2)S_i^2S_j^2 + (1-\varphi _{jk}^2)S_j^2S_k^2 + (1-\varphi _{ik}^2)S_i^2S_k^2 \\ & \quad + (2\varphi _{ij} + 2\varphi _{jk}\varphi _{ik})C_iC_jS_iS_jS_k^2 + (2\varphi _{jk} + 2\varphi _{ij}\varphi _{ik})C_jC_kS_jS_kS_i^2 + (2\varphi _{ik} + 2\varphi _{ij}\varphi _{jk})C_iC_kS_iS_kS_j^2] \\ & =\frac{1}{32}\Big\{ & c\Big[x^2y^2z^2-2x^2y^2z+x^2y^2-2x^2yz^2+4x^2yz-2x^2y+x^2z^2-2x^2z+x^2\\[1mm]& \quad -2xy^2z^2+4xy^2z-2xy^2+4xyz^2-8xyz+4xy-2xz^2+4xz-2x\\[1mm]& \quad +y^2z^2-2y^2z+y^2-2yz^2+4yz-2y+z^2-2z+1\Big]\\[1mm]& +2a_1\Big[x^2y^2-2x^2y-2xy^2+4xy+x^2+y^2-2x-2y+1\Big]\\[1mm]& +2a_2\Big[x^2z^2-2x^2z-2xz^2+4xz+x^2+z^2-2x-2z+1\Big]\\[1mm]& +2a_3\Big[y^2z^2-2y^2z-2yz^2+4yz+y^2+z^2-2y-2z+1\Big]\\[1mm]& +b_1\Big[x^2y^2z^2-2x^2y^2z+x^2y^2-x^2z^2+2x^2z-x^2-y^2z^2+2y^2z-y^2 +z^2-2z+1\Big]\\[1mm]& +b_3\Big[x^2y^2z^2-2xy^2z^2+y^2z^2-x^2y^2+2xy^2-y^2-x^2z^2+2xz^2-z^2+x^2-2x+1\Big]\\[1mm]& +b_2\Big[x^2y^2z^2-2x^2yz^2+x^2z^2-x^2y^2+2x^2y-x^2-y^2z^2+2yz^2-z^2+y^2-2y+1\Big] \Big\} . \end{aligned}\end{equation*}