Dealing with multicollinearity by separating common variance

#1
On this page, I read about multiple ways to deal with multicollinearity: http://www.chsbs.cmich.edu/fattah/courses/empirical/multicollinearity.html. I determined that the last method suggested may be best for my purposes. I copied it below:

"Treat the common variance as a separate variable and decontaminate each covariate by regressing them on the others and using the residuals. That is, analyze the common variance as a separate variable. "

I think that I get this conceptually but do not know how to do it in SPSS or R. If someone could explain how to do it in SPSS (preferably) or R, I'd appreciate it!