Symmetrisation and the Feigin-Frenkel centre

For simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the Feigin-Frenkel centre. These formulas make use of the symmetrisation map as well as some well-chosen symmetric invariants of $\mathfrak g$. There are some general results on the role of the symmetrisation map in the explicit description of the FF-centre. The advantage of our method is that it reduces questions about elements of the FF-centre to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G$_2$ by hand.