Weak Bott periodicity vs. strong Bott periodicity

This has been asked and answered on MathOverflow. I have replicated the accepted answer by Peter May below.

As a matter of history, the original Bott maps are very explicit, and in fact they are $E_{\infty}$ maps with respect to the actions of the linear isometries operad (as shown in the first chapter of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra). Bott himself, in his paper Raoul Bott. Quelques remarques sur les théorèmes de périodicité. Bull. Soc. Math. France 87 1959 293–310, showed how to derive the strong form from the weak form by showing that his original maps are homotopic to the adjoints of the evident maps obtained by tensoring with the Bott classes. He does this for the real and quaternionic cases as well.