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Abstract: Recall that a structure (group, Lie algebra, associative algebra, etc) M is omega-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is omega-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M) on the cartesian power M^n, for all natural number n. In 1981, Wilson conjectured that any omega-categorical locally nilpotent group is nilpotent. If true, a quite satisfactory decomposition of omega-categorical groups would follow. This conjecture is very much open more than 40 years later. The analogue statement for Lie algebras (every locally nilpotent omega-categorical Lie algebra is nilpotent) is also open and, as it turns out, it reduces to proving that for each n and prime p, every omega-categorical n-Engel Lie algebra over F_p is nilpotent. As for associative algebras, the analogous question was already answered by Cherlin in 1980: every locally nilpotent omega-categorical ring is nilpotent. We see the Wilson conjecture for Lie algebra as a bridge between the result of Cherlin and the original question of Wilson for omega-categorical groups. The question of Wilson, for groups, for Lie algebras or for associative algebras are connected to classical nilpotency problems such as the Burnside problem, the Kurosh problem or the problem of local nilpotency of n-Engel groups. Using a classical result of Zelmanov, the Wilson conjecture for omega-categorical Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over F_p is nilpotent for all but finitely many p's. The situation for small values of the pair (n,p) is as follows: . Every 2-Engel Lie algebra is nilpotent (Higgins 1954), . Every 3-Engel Lie algebra over F_p with p\neq 2,5 is nilpotent (Higgins 1954), . Every 4-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5. (Higgins 1954, Kostrikin 1959), . Every 5-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5,7 (Vaughan-Lee, 2024). In other words, for (n,p) = (3,2), (3,5), (4,2), (4,3), (4,5),... It is known that n-Engel Lie algebras of char p are not globally nilpotent. Our goal, on the long run, is to prove that for those values of (n,p), omega-categorical n-Engel Lie algebra of characteristic p are nilpotent. We have recently dealt with the cases (n,p) = (3,5) and (n,p) = (4,3), and the proofs are different both in taste and method. The goal of the talk is to present a proof that every omega-categorical 4-Engel Lie algebras of characteristic 3 is nilpotent. Our solution of the case at hand consists in adapting in the definable context some classical tools for studying Engel Lie algebras, appearing earlier in the work of Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others. Our solution involves the use of computer algebra.