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We construct a moduli space   $\textsf {LP}_{G}$   of   $\operatorname {SL}_{2}$  -parameters over   ${\mathbb {Q}}$  , and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a  Jacobson–Morozov morphism    $\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$   (where   $\textsf {WDP}_{G}$   is the moduli space of Weil–Deligne parameters considered by several other authors). We show that   $\textsf {JM}$   is an isomorphism over a dense open of   $\textsf {WDP}_{G}$  , that it induces an isomorphism between the discrete loci   $\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$  , and that for any   ${\mathbb {Q}}$  -algebra   $A$   it induces a bijection between Frobenius semi-simple equivalence classes in   $\textsf {LP}_{G}(A)$   and Frobenius semi-simple equivalence classes in   $\textsf {WDP}_{G}(A)$   with constant (up to conjugacy) monodromy operator.

The Jacobson–Morozov Morphism for Langlands Parameters in the Relative Setting