Cartan Invariants of Reduced Enveloping Algebras of Cartan Type

Let k be an algebraically closed field of characteristic p>3 and let g be a restricted simple Lie algebra of Cartan type W, S, H, or K. We determine the complete labelled Cartan matrix of u_chi(g) for every p-character chi. At zero character the Witt, Special, Hamiltonian, and stable-range Contact matrices are given by outer-product formulas; the remaining Contact cases are extracted from a full finite corner computed from restricted-PBW data. The nonzero height-zero Witt and Special matrices are principal deletions of their zero-character matrices. For every other nonzero character, a family-wise normalization gives a matrix factor u_chi(g) = Mp^r(C), r >= 1, where C has an explicit basis obtained by projecting the restricted-PBW basis and row-reducing, together with an explicit multiplication table. Primitive decomposition of this corner gives [P_alpha : L_beta] = dim e_beta C e_alpha and hence the full Cartan matrix. This solves Problem 1 of Benkart and Feldvoss for the four Cartan families in the stated characteristic range. We also give finite-W refinements for the classical type-A and type-C sources and recover the Feldvoss-Nakano rank-one Witt tables.