On an Algebraic Structure of Dimensionally Reduced Magical Supergravity Theories

We study an algebraic structure of magical supergravities in three dimensions. We show that if the commutation relations among the generators of the quasi-conformal group in the super-Ehlers decomposition are in a particular form, then one can always find a parameterization of the group element in terms of various 3d bosonic fields that reproduces the 3d reduced Lagrangian of the corresponding magical supergravity. This provides a unified treatment of all the magical supergravity theories in finding explicit relations between the 3d dimensionally reduced Lagrangians and particular coset nonlinear sigma models. We also verify that the commutation relations of $E_{6(+2)}$, the quasi-conformal group for $\mathbb{A}=\mathbb{C}$, indeed satisfy this property, allowing the algebraic interpretation of the structure constants and scalar field functions as was done in the $F_{4(+4)}$ magical supergravity.