Nonassociative enveloping algebras of Chevalley algebras

An algebra $R$ is said to be an exact enveloping algebra for a Lie algebra $L$ if $L$ is isomorphic to the algebra $R^{(-)}$ obtained by replacing the multiplication in $R$ by the commutation: $a*b:= ab- ba$. We study exact enveloping algebras of certain subalgebras of a Chevalley algebra over a field $K$ associated with an indecomposable root system $\Phi$. The structure constants of the Chevalley basis of this algebra are chosen with a certain arbitrariness for the niltriangular subalgebra $N\Phi(K)$ with the basis $\{e_r\ |\ r\in\Phi^+\}$. The exact enveloping algebras $R$ for $N\Phi(K)$, which were found in 2018, depend on this choice. The notion of standard enveloping algebra is introduced. For the type $A_{n-1}$, one of the exact enveloping algebras $R$ is the algebra $NT(n,K)$ of all niltriangular $n\times n$ matrices over $K$. The theorem of R. Dubish and S. Perlis on the ideals of $NT(n,K)$ states that $R$ is standard in this case. We prove that an associative exact enveloping algebra $R$ of a Lie algebra $NT(n,K)$ of type $A_{n-1}$ $(n>3)$ is unique and isomorphic to $NT(n,K)$ up to passing to the opposite algebra $R^{({\rm op})}$. Standard enveloping algebras $R$ are described. The existence of a standard enveloping algebra is proved for the Lie algebras $N\Phi(K)$ of all types excepting $D_{n}$ $(n\geq 4)$ and $E_{n}$ $(n=6,7,8)$.