  A simple algorithm, which exploits the associativity of the BCH formula, and
that can be generalized by iteration, extends the remarkable simplification of
the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and
Visser. We show that if
  $[X,Y]=uX+vY+cI$, $[Y,Z]=wY+zZ+dI$, and, consistently with the Jacobi
identity, $[X,Z]=mX+nY+pZ+eI$, then $$
\exp(X)\exp(Y)\exp(Z)=\exp({aX+bY+cZ+dI}) $$ where $a$, $b$, $c$ and $d$ are
solutions of four equations. In particular, the Van-Brunt and Visser formula
$$\exp(X)\exp(Z)=\exp({aX+bZ+c[X,Z]+dI}) $$ extends to cases when $[X,Z]$
contains also elements different from $X$ and $Z$. Such a closed form of the
BCH formula may have interesting applications both in mathematics and physics.
As an application, we provide the closed form of the BCH formula in the case of
the exponentiation of the Virasoro algebra, with ${\rm SL}_2({\rm C})$
following as a subcase. We also determine three-dimensional subalgebras of the
Virasoro algebra satisfying the Van-Brunt and Visser condition. It turns out
that the exponential form of ${\rm SL}_2({\rm C})$ has a nice representation in
terms of its eigenvalues and of the
  fixed points of the corresponding M\"obius transformation. This may have
applications in Uniformization theory and Conformal Field Theories.
