Canonical isometric embeddings of projective spaces into spheres

We define inductively isometric embeddings of $\mb{P}^n(\mb{R})$ and $\mb{P}^n(\mb{C})$ (with their canonical metrics conveniently scaled) into the standard unit sphere, which present the former as the restriction of the latter to the set of real points. Our argument parallels the telescopic construction of $\mb{P}^\infty(\mb{R})$, $\mb{P}^\infty(\mb{C})$, and $\mb{S}^\infty$ in that, for each $n$, it extends the previous embedding to the attaching cell, which after a suitable renormalization makes it possible for the result to have image in the unit sphere.