Gauss-Manin Lie algebra of mirror elliptic K3 surfaces

We study mirror symmetry of families elliptic K3 surfaces with fibers type $E_6,~E_7$ and $E_8$. consider a moduli space $\mathsf{T}$ the enhanced choice differential forms. show that coordinates on are given by ring quasi modular forms in two variables, groups adapted to fiber type. furthermore introduce an algebraic group $\mathsf{G}$ which acts from right construct its Lie algebra $\mathrm{Lie}(\mathsf{G})$. prove extended generated $\mathrm{Lie}(\mathsf{G})$ together vector fields is isomorphic $\mathrm{sl}_2(\mathbb{C})\oplus\mathrm{sl}_2(\mathbb{C})$.