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We extend results of Looijenga--Lunts and Verbitsky and show that the total Lie algebra $$\mathfrak g$$ for the intersection cohomology of a primitive symplectic variety $$X$$ with isolated singularities is isomorphic to $$\mathfrak g \cong \mathfrak{so}\left(\left(IH^2(X, \mathbb Q), Q_X\right)\oplus \mathfrak h\right),$$ where $$Q_X$$ is the intersection Beauville--Bogomolov--Fujiki form and $$\mathfrak h$$ is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperk\"ahler metric. Along the way, we study the structure of $$IH^*(X, \mathbb Q)$$ as a $$\mathfrak{g}$$-representation -- with particular emphasis on the Verbitsky component, multidimensional Kuga--Satake constructions, and Mumford--Tate algebras -- and give some immediate applications concerning the $P = W$ conjecture for primitive symplectic varieties. Comment: 41 pages; Final journal version; new subsection on LLV algebra for symplectic orbifolds