2x2x3 ANOVA interaction and error term formulas?

Hey guys, I would really appreciate some help with some ANOVA formulas. :)

So I'm running a 2 (between) by 2 (between) by 3 (within) design and I want to run some ANOVAs by hand to (a) double-check the SPSS output and (b) have some more control over which error term is used. Really, I have four groups that I measure three times, but my committee told me to break those four groups down into a 2x2, so that's how it goes.

I'm fine up with calculating split-plot ANOVAs with a single between and a single within variable, but I'm not sure how it extends when you have a second between-variable like I do.

So, in the notation I learned, the sum of square formula for a split-plot between-variable (A) with a levels (with a within-variable B with b levels), you use:
View attachment 3425

For the within-variable, the SS formula would be:
View attachment 3426

The interaction of AxB would be:
View attachment 3427

and the within-error term used for the main effect of B and the AxB interaction would be:
View attachment 3428

But how would this extend to the three-way interaction (or even the between-interaction with two between variables)? :confused:

Again, I would really appreciate some input! Thanks! :)