Enveloping algebras and ideals of the niltriangular subalgebra of the Chevalley algebra

A simple complex Lie algebra is characterized by a root system $\Phi$ and a Chevalley basis with the integer structure constants. The well-known arbitrariness of their choice for the niltriangular subalgebra $N\Phi(C)$ essentially affects the Lie-admissible algebra $R_\Phi$ (in the sense of Albert) over a field $K$ such that $R_\Phi^{(-)}\simeq N\Phi(K)$. We study the uniqueness of the (nonassociative) enveloping algebras $R_\Phi$ of classical types. The enumeration of ideals of the Lie algebras $N\Phi(K)$ and $R_\Phi$ for $K=GF(q)$ leads to the solution of some combinatorial problem listed in ACM SIGSAM Bulletin in 2001. The calculations of multiple combinatorial sums with $q$-binomial coefficient use the integral representation method of combinatorial sums (the coefficient method).