  This paper investigates the simultaneous twisting of the Courant bracket by a
2-form $B$ and a bi-vector $\theta$, exploring the generalized fluxes obtained
in Courant algebroid relations. We define the twisted Lie bracket and
demonstrate that the generalized $H$-flux can be expressed as the field
strength defined on this Lie algebroid. Similarly, we show that the $f$-flux is
a direct consequence of simultaneous twisting, which arises in the twisted Lie
bracket relations between holonomic partial derivatives. We obtain the
generalized $Q$ flux in terms of the twisted Koszul bracket, which is a
quasi-Lie algebroid bracket. The action of an exterior derivative related to
the twisted Koszul bracket on a bi-vector produces the generalized $R$-flux. We
show that the generalized $R$-flux is also the twisted Schouten-Nijenhuis
bracket, i.e. the natural graded bracket on multi-vectors defined with the
twisted Lie bracket.
