Standard inclusions in the Cayley–Dickson tower

Statement

The standard Cayley–Dickson tower gives injective unital real-algebra maps

\mathbb R\hookrightarrow\mathbb C\hookrightarrow\mathbb H \hookrightarrow\mathbb O\hookrightarrow\mathbb S.

At each doubling, the inclusion is $a\mapsto(a,0)$ . Here the last object is the real sedenion algebra; all arrows are interpreted in the category of real algebras with no associativity requirement.

Verification

In a Cayley–Dickson product, $(a,0)(b,0)=(ab,0)$ . This proves product preservation; linearity, preservation of one, and injectivity follow from the coordinates. Standard conjugation also restricts correctly. The inclusion of $\mathbb C$ in $\mathbb H$ uses the chosen quaternion generator $i$ .

Category boundary

The first three objects are associative; the octonions and sedenions are not. The final arrow cannot be an arrow in the category of composition algebras, since sedenions do not have a multiplicative nondegenerate quadratic norm. Likewise, choosing the complex subalgebra inside $\mathbb H$ does not make quaternion multiplication complex-bilinear.