3D projection of a hyperellispoid in 6D

Given a $6\times 6$ real spd matrix $A$, the equation $$x^\top A x=1$$ defines a hyperellipsoid.

I would like to represent the projection of this ellipsoid in a 3D space, namely $(x_1,x_2,x_4)$. I eliminated one of the variables using the above equation, then numerically I can generate the points in 6D and select their $(1,2,4)$ components, but I believe there is a more efficient approach...

Is there a simple way to do that?

I finally managed. First perform an orthogonal projection to $(x_1,x_2,0,x_4,0,0)$. The gradient of the surface is orthogonal the projection directions, providing 3 equations:

$$e_3^\top A x = 0, \quad e_5^\top A x = 0,\quad e_6^\top A x = 0,$$ to be completed by $x^\top A x=0$. This provides a 2-dimensional manifold, which can then be plotted in 3D.