  We consider the infinitesimal freeness in the operator-valued framework, and
we show that the operator-valued infinitesimal (OVI) free independence is
equivalent to the operator-valued free independence over an algebra of $2\times
2$ upper triangular matrices. We introduce the notion of OVI cumulants and
investigate its properties, and we then deduce that the OVI freeness is
equivalent to the vanishing of our mixed cumulants. Moreover, we derive the
formula for obtaining the free additive and multiplicative convolutions within
the realm of OVI freeness.
