I asked this on MSE over a month ago, but the one answer I got doesn't seem to work.
Let $\times$ denote the cross-product. $\;$ Is it the case that
For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in $\mathbf{R}^3$, $\;\;\;\; \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{z}\hspace{.02 in}\right|\right| \:\: \leq \:\: \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{y}\hspace{.02 in}\right|\right| \hspace{.02 in}+\hspace{.02 in}\left|\left|\hspace{.03 in}\mathbf{y} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{z}\hspace{.02 in}\right|\right| \;\;\;\;\;$.
?
(If yes, then $\;\;\; \langle [\mathbf{x}]\hspace{.01 in},\hspace{-0.03 in}[\hspace{.02 in}\mathbf{y}] \rangle \: \mapsto \: \left|\left|\hspace{.03 in}\mathbf{x} \hspace{-0.03 in}\times \hspace{-0.03 in}\mathbf{y}\hspace{.02 in}\right|\right| \;\;\;$ defines a nice metric on the projective plane.)