Semisimplification of contragredient Lie algebras

We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator.In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra.This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra.We also obtain a non-degenerate invariant form that behaves well with the lattice grading.As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category.