  The splitting of $\e^{h(A+B)}$ into a single product of $\e^{h A}$ and
$\e^{hB}$ results in symplectic integrators when $A$ and $B$ are classical Lie
operators. However, at high orders, a single product splitting, with
exponentially growing number of operators, is very difficult to derive. This
work shows that, if the splitting is generalized to a sum of products, then a
simple choice of the basis product reduces the problem to that of
extrapolation, with analytically known coefficients and only quadratically
growing number of operators. When a multi-product splitting is applied to
classical Hamiltonian systems, the resulting algorithm is no longer symplectic
but is of the Runge-Kutta-Nystr\"om (RKN) type. Multi-product splitting, in
conjunction with a special force-reduction process,explains why at orders $p=4$
and 6, RKN integrators only need $p-1$ force evaluations.
