(Last update: 2/19/16)

  • Construction of $\OO$

Using the Cayley-Dickson process of §5.1, $\OO$ is constructed from $\HH$ merely by specifying that $\ell^2=-1$; the rest of the multiplication table follows from the Cayley-Dickson product. If instead one merely specifies $\ell^2=-1$, then further assumptions are needed in order to recover the full multiplication table.

One possibility is to assume two independent imaginary units determine a quaternionic subalgebra, which is enough to show that $i\ell$, $j\ell$, and $k\ell$ must also square to $-1$, but still not quite enough to determine the products of these elements with each other.

  • $\Sp(2)\cong\Spin(5)$

It is shown in §8.2 that $\Sp(4,\RR)\cong\Spin(3,2)$, but it is later asserted in §9.4 without proof that $\Sp(2)\cong\Spin(5)$. It is enough to note that these latter groups are the compact forms of the former, but a more explicit construction can also be made: Replace $q$ and $t$ in (8.10) by $iq$ and $it$, respectively, then argue that the condition $M^\dagger M=I$ from (8.6) forces the Lie algebra elements to be anti-Hermitian, and hence rotations rather than boosts, with compact orbits. Or argue that such $M$ preserve the new form of $P$, in analogy with the argument on p.59.

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