Argument for symmetry

Question

There is this question regarding constrained optimisation. It says, a rectangular parallelepiped has all eight vertices on the ellipsoid $x^{2}+3y^{2}+3z^{2}=1$. Using the symmetry about each of the planes, write down the surface area of the parallelepiped and therefore find the maximum surface area.
I know that the surface area $S(x,y,z) = 8xy+8yz+8zx$. I've also defined the constraint as given $g(x) = x^{2}+3y^{2}+3z^{2}=1$. Using the Lagrange multiplier $\lambda$ I got:
$8y+8z-2\lambda x=0$
$8x+8z-6\lambda y=0$
$8x+8y-6\lambda z=0$
What confuses me is what is next in the answer key. They say using symmetry, $y=z$. My question is, how are we able to make that argument? I'm slightly confused. Is it because in the ellipsoid equation, the coefficients of $y$ and $z$ are equal? Does that mean for a rectangular parallelepiped inscribed in said ellipsoid, the $y$ and $z$ coordinates of its vertices will always be equal? If this is the case, can anyone explain to me why?

Answer 1

Subtracting one from the other the last two equations one gets: $$ (8+6\lambda)(z-y)=0. $$ Hence either $\lambda=-4/3$ (which would lead to $y=-z$ and is thus to discard) or $y=z$.

Answer 2

Even before using the method of Lagrange multipliers, we may note exchanging $y$ with $z$ preserves the Lagrangian, so the solution will have $y=z$.