  In this paper, we first introduce the concept of symmetric biderivation
radicals and characteristic subalgebras of Lie algebras, and study their
properties. Based on these results, we precisely determine biderivations of
some Lie algebras including finite-dimensional simple Lie algebras over
arbitrary fields of characteristic not $2$ or $3$, and the Witt algebras
$\mathcal{W}^+_n$ over fields of characteristic $0$. As an application,
commutative post-Lie algebra structure on aforementioned Lie algebras is shown
to be trivial.
