 Solution

The symmetric part of the displayed matrix isIt equals the matrix of 8 plus $3I$, so the same orthonormal eigenbasis gives eigenvalues $6,1,1$. In coordinates $u,v,w$ along ${\mathbf{v}}_3,{\mathbf{v}}_{-2},{\mathbf{e}}_2$, respectively, the level set isAll three eigenvalues are positive, so this is an ellipsoid with semi-axis lengths $1,\sqrt{6},\sqrt{6}$. The short axis is along $(2,0,1)$, and the plane of the longest axes is $2x_1+x_3=0$.

A natural longest-axis direction in the $x_1{x}_3$ plane is ${\mathbf{v}}_{-2}=(1,0,-2)/\sqrt{5}$. The angle $\eta$ of a unit vector to the plane $x_1=0$ has $\sin \eta =|v_1|$, givingfor that choice. Degeneracy qualification: the two longest semi-axes are equal, so “the longest axis” does not specify a unique direction. The equally valid longest axis ${\mathbf{e}}_2$ lies in $x_1=0$ and has angle zero. Generally $\mathbf{v}=\cos \psi {\mathbf{v}}_{-2}+\sin \psi {\mathbf{e}}_2$ is a longest-axis unit vector, withThis is the principal-axis freedom in a repeated ellipsoid eigenspace; a unique answer needs an extra axis-selection convention.