# Why can't I use just two angles when considering three neutrino oscillation?

I'm just finishing basic particle physics and I just learned about neutrino oscillation. Considering only two neutrinos, one can describe neutrino "mass states" (as we call it) with an angle $$\theta$$ like so:
$$\pmatrix{|\nu_1\rangle\\|\nu_2\rangle}=\pmatrix{\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta}\pmatrix{|\nu_e\rangle\\|\nu_\mu\rangle}.$$
We can do so if we ignore complex phase (which by itself can't be measured) and if we use the fact, that the wave function must be normalized. Essentially we are, of course, describing a 1-sphere, where each point of its surface describes unique combination of $$|\nu_e\rangle$$ and $$|\nu_\mu\rangle$$ such that the square of the magnitude of this superposition is always 1. All is good.
The weird thing is that the three neutrino oscilation requires three angles $$\theta_{12}$$ $$\theta_{23}$$ and $$\theta_{13}$$. Why? Isn't a 2-sphere good enough to describe all possible normalized superpositions of three states? Like this it seems to describe one extra rotation as such:

but that doesn't change the linear combinantion of the states!
Please don't respond with the derivation of the 3×3 matrix, that has been unhelpful so far.

In the abstract vector space of neutrino physics, there are two coordinate systems, which you should think of like the $$\vec x, \vec y, \vec z$$ coordinate system in your figure. One is the flavor basis: a set of vectors $$\nu_e, \nu_\mu, \nu_\tau$$ at right angles to each other. The other is the mass basis: a set of not-flavors $$\nu_\text{light}, \nu_\text{middle}, \nu_\text{heavy}$$ which are also at right angles to each other. To describe the orientation of one set of axes in terms of the other takes three rotations; with two angles you can get your head above the water but you might be stuck facing away from the beach.