It is well known that on $\mathbb{R}P^2$ one can define the following ''cross product metric'' $$d([x],[y]) = \frac{\|x \times y\|}{\|x\| \|y\|},$$ where $x,y \in \mathbb{R}^3 \setminus \{0\}$ and $\|.\|$ is the standard Euclidean norm. But let us deform it with the help of a 3-by-3 nonsingular matrix $M$ as follows $$d_M([x],[y]) = \frac{\|M(x \times y)\|}{\|x\| \|y\|}.$$ By the polar decomposition, we can assume $M$ is positive-definite. The question is: For which such $M$'s is $d_M$ a metric?
The only issue is, of course, the triangle inequality. It is not hard to find the following necessary condition: the eigenvalues $\lambda_1,\lambda_2,\lambda_3$ of $M$ must themselves satisfy the triangle inequalities $$\lambda_1 \leq \lambda_2 + \lambda_3, \quad \lambda_2 \leq \lambda_3 + \lambda_1, \quad \lambda_3 \leq \lambda_1 + \lambda_2,$$ what can be neatly expressed as $\textrm{tr}M \geq 2\rho(M)$, where $\rho(M)$ is the spectral radius of $M$. Numerics seems to suggest that this condition is also sufficient, but I haven't been able to prove that (apart from the trivial case $\lambda_1=\lambda_2=\lambda_3$).
I've tried taking $\|x\| = \|y\| = \|z\| = 1$ and expressing $d_M([x],[z]) + d_M([z],[y]) - d_M([x],[y])$ in spherical coordinates, but everything becomes hopelessly complicated. I also know $d_M$ can be equivalently written as: $$d_M([x],[y]) = \frac{\|M^{-1}x \times M^{-1}y\|}{\|x\| \|y\|} \det M,$$ but this doesn't help much either. I would be very grateful for any hints and/or references.