Gröbner–Shirshov bases of the Lie algebra $B_n^+$

The minimal Gröbner–Shirshov bases of the positive part $B_n^+$ of the simple finite-dimensional Lie algebra $B_n$ over an arbitrary field of characteristic $0$ are calculated, for the generators associated with simple roots and for an arbitrary ordering of these generators (i.e., an arbitrary one of $n!$ Gröbner–Shirshov bases is chosen and studied). This is a completely new class of problems; till now this program was carried out only for the Lie algebra $A_n^+$. The minimal Gröbner–Shirshov basis of the Lie algebra $B_n^+$ was calculated earlier by Bokut and Klein, but this was done for only one ordering of generators.