Splitting the Bézier cubic defined by the four control points $P, Q, R$, and $S$ in two parts corresponding to the two parametric subintervals $[0, t]$ and $[t, 1]$ is relatively easy: we perform linear interpolations that result in new control points $P, PQ, PQR, PQRS$ and $PQRS, QRS, RS, S$.
Now I want to generalize to the case of the three subintervals $[0,t], [t, u]$ and $[u,1]$ in a single go. The control points for the left and right arcs are found as above, using the fractions $t$ and $u$, i.e. $P,PQ_t,PQR_t,PQRS_t$ and $PQRS_u,QRS_u, RS_u, S$. What I am missing is how to compute the two remaining control points for the middle arc.
I know that I could split as $[0,t]$ and $[t,1]$, then split the second subinterval using $[0,\frac u{1-t}]$ and $[\frac u{1-t}]$, but I think/hope that a direct approach is possible.
