Gauss-Manin Lie algebra of mirror elliptic K3 surfaces

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