Question about term in Oasisx splitting scheme

In this webpage a splitting scheme for the Navier Stokes equations is discussed. It is first discussed for the Stokes equation, then for the Navier Stokes equations.

If I repeat step by step the procedure for the Stokes equations, I obtain an additional term that is not present in the equations at the beginning of Section " Navier-Stokes equation".

In fact, the equations for {\bf u}^n are

\frac{ \bf u^n - u^{n-1}}{\Delta t} + \left( \frac{3}{2} { \bf u}^{n-1} - \frac{1}{2} { \bf u}^{n-2}\right) \cdot \nabla \left[ \frac{1}{2} \left( \bf u^{n} + u^{n-1}\right) \right] + \cdots = f^{n-1/2}

and those for \bf u^\ast are

\frac{ \bf u^\ast - u^{n-1}}{\Delta t} + \left( \frac{3}{2} { \bf u}^{n-1} - \frac{1}{2} { \bf u}^{n-2}\right) \cdot \nabla \left[ \frac{1}{2} \left( \bf u^{\ast} + u^{n-1}\right) \right] + \cdots = f^{n-1/2}.

Following the procedure of Section “Stokes equations,” I sustract the two equations above and take the divergence. I obtain

\nabla^2 \phi = \left[ \frac{1}{\Delta t} + \frac{1}{2} \left( \frac{3}{2} { \bf u}^{n-1} - \frac{1}{2} { \bf u}^{n-2}\right) \right] (\nabla \cdot \bf u^\ast),

which differs from the fourth equation in Section “Navier Stokes equations.” (the - sign in the right-hand side in the latter equation is wrong, but that is a different issue, let us forget about that for a second).

What I would like to understand is why in my equation there is the additional term

\frac{1}{2} \left( \frac{3}{2} { \bf u}^{n-1} - \frac{1}{2} { \bf u}^{n-2}\right)

while in the webpage there is no such term.
Has this term been neglected here because it is small compared to 1/\Delta t?

Here there seems to be several steps missing, as you need to take the divergence of the \bf u \cdot \nabla \bf u term, which is missing.