C02.2 · Definitions/conventions

In the matrix convention of C02.1, \[ X=\begin{pmatrix}a&z&\bar y\\\bar z&b&x\\y&\bar x&c\end{pmatrix}. \] Let \(E_i\) be the diagonal matrix idempotents. The three octonionic coordinate subspaces are \(J_{12}\) for \(z\), \(J_{13}\) for \(y\), and \(J_{23}\) for \(x\). Thus \[ J_3(\mathbb O_s)= \mathbb R E_1\oplus\mathbb R E_2\oplus\mathbb R E_3 \oplus J_{12}\oplus J_{13}\oplus J_{23}. \] The dimensions are computed in `Peirce.dimensions`, not used as a family-replication argument.

On the diagonal algebra, distinguish the homogeneous map \[ \widetilde T:\mathbb R^3\longrightarrow\mathbb R^3,\qquad \widetilde T(x,y,z)=(yz,xz,xy) \] from its restriction \(T=\widetilde T|_{S^2}\), with \(x^2+y^2+z^2=1\). The latter is the compact parametrized Roman surface of the source. A polynomial equation for its image is not automatically a complete description of its real image.