The following theorem comes from the 7th edition of "Introduction to Mathematical Statistics" by Hogg, Craig and Mckean and it concerns the necessary and sufficient condition for the independence of two quadratic forms of normal variables.
This is a rather long extract but what I would appreciate some help with is only the transition from 9.9.6 to 9.9.7. I have included the previous steps just to give the overall picture in case a previous result is implicitly used. Could you please help me understand then why 9.9.6 and 9.9.7 are equivalent representations? I have tried deriving 9.9.7 on my own but all my attempts ended in frustration.
The proof goes on after that but I do not have any other problems. Thank you in advance.