Identifying models of the octave projective plane (Q1357635)

Let \(\pi\) be the projective plane over the alternative semifield \({\mathcal O}\) of Cayley numbers (called octaves in the article). \textit{H. Aslaksen} [Geom. Dedicata 40, No. 2, 245-250 (1991; Zbl 0758.51001)] used a modified homogeneous coordinate system for representing \(\pi\). Aslaksen identified the points of \(\pi\) with the nonzero triples \((x_1, x_2, x_3)\) with each \(x_i\in {\mathcal O}\) and at least one real. Furthermore, \((x_1, x_2, x_3)= (y_1, y_2, y_3)\) if and only if there exists \(a\in {\mathcal O}\) such that \(y_i=ax_i\) for each \(i\). Earlier, \textit{H. Freudenthal} [Geom. Dedicata 19, 7-63 (1985; Zbl 0573.51004)] identified the points of \(\pi\) with the set of idempotents in the exceptional Jordan algebra \({\mathcal J}\) consisting of the 3 by 3 hermetian matrices over \({\mathcal O}\) and having the multiplication \(A\times B= (AB+ BA)/2\). This article shows how to go from Aslaksen's representation to Freudenthal's by the mapping \(\alpha\): \((x_1, x_2, x_3) \to[e_{ij}]\) with \(e_{ij}= \overline {x_i}x_j\). The proof that \(\alpha\) is an isomorphism depends upon a lemma first given by \textit{P. Jordan} [Abh. Math. Semin. Univ. Hamb. 16, No. 1/2, 74-76 (1949; Zbl 0034.38103)].