there are precisely 3 such quaternionic subalgebras of the octonions, which we interpret as generations.

There is a long history of trying to use the quaternions in 4-dimensional quantum mechanics; see the comprehensive treatment in [@Adler] and references therein. Our approach is different in that we use the additional degrees of freedom to repackage existing information, rather than increasing the size of the underlying space of scalars. Ultimately, this leads us to work in more than 4 spacetime dimensions.

We also note a relatively unknown paper by Dirac [@Dirac] which, much to our surprise, contains the precursors of several of the key ideas presented here.

The octonions were first introduced into quantum mechanics by Jordan [@Jordan; @JNW]. There has in fact been much recent interest in using the octonions in (higher-dimensional) field theory; excellent modern treatments can be found in [@GT; @Okubo; @Dixon].

After much of this work was completed, we became aware of the recent work of Schücking [*et al.*]{} [@Schucking; @SubStandard], who also use a quaternionic formalism to describe a single generation of leptons. They further speculate that extending the formalism to the octonions would yield a description of a single generation of quarks as well. Although the 