mutliple interactions in same equation

#1
Hi Folks,

I am reviewing a manuscript that uses four interactions in the same equation, and i am wondering if authors need to include higher order terms. Here is their basic additive model (with other predictors dropped for simplicity):

y= a + x1 + x2 + x3 + x4

They are testing for interatcions between (x1 & x3) and (x1 & x4) and (x2 & x3) and (x2 & x4). They test for these interactions in a single model, using this equation:

y= a + x1 + x2 + x3 + x4 + x1x3 + x1x4 + x2x3 + x2x4

Is this correct? Are they missing any other (higher order) cross-products? I am concerned that x3 and x4 are distributed across two variables, thus implying a three-way interaction.

Any quick suggestions you could offer would be appreciated.

Thanks in advance,
Bill