  The central problem in this technical report is the question if the classical
Bernstein operator can be decomposed into nontrivial building blocks where one
of the factors is the genuine Beta operator introduced by M\"uhlbach and
Lupa\c{s}. We collect several properties of the Beta operator such as
injectivity, the eigenstructure and the images of the monomials under its
inverse. Moreover, we give a decomposition of the form $B_n =
\bar{\mathbb{B}}_n \circ F_n $ where $F_n$ is a nonpositive linear operator
having quite interesting properties. We study the images of the monomials under
$F_n$, its moments and various representations. Also an asymptotic formula of
Voronovskaya type for polynomials is given and a connection with a conjecture
of Cooper and Waldron is established. In an appendix numerous examples
illustrate the approximation behaviour of $F_n$ in comparison to $B_n$.
