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The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proofof the norm convergence in $L^2(\mu)$ of the quadratic nonconventional ergodicaverages \frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circT_1^{n^2}T_2^n)\quad\quad f_1,f_2\in L^\infty(\mu) associated to an arbitraryprobability-preserving \bbZ^2-system (X,\mu,T_1,T_2). This is a special case ofthe Bergelson-Leibman conjecture on the norm convergence of polynomialnonconventional ergodic averages.  That proof relied on some new machinery for extending probability-preserving$\bbZ^d$-systems to obtain simplified asymptotic behaviour for variousnonconventional averages such as the above. The engine of this machinery isformed by some detailed structure theorems for the `characteristic factors'that are available for some such averages after ascending to asuitably-extended system. However, these new structure theorems underwent twodistinct phases of development, separated by the discovery of some newtechnical results in Moore's cohomology theory for locally compact groups(arXiv:1004.4937). That discovery enabled a significant improvement to the mainstructure theorem (Theorem 1.1 in (arXiv:0905.0518)), which in turn afforded amuch shortened proof of convergence. However, since the proof of convergenceusing the original structure theorem required some quite different ideas thatare now absent from these other papers, I have recorded it here in case it hassome independent interest.

An alternative ending to "Pleasant extensions retaining algebraic structure''