= Solution
At the specified state, the <symmetric matrix> is
$$
S=\begin{pmatrix}-20&12&0\\12&-2&0\\0&0&-6\end{pmatrix}.
$$
The planar characteristic equation is $\lambda^2+22\lambda-104=0$, giving $4$ and $-26$, while the third <eigenvalue> is $-6$. An orthonormal set of <Eigenvector>[eigenvectors] is
$$
\boxed{\lambda_1=4,\quad\mathbf e_1=\frac{(1,2,0)^T}{\sqrt5};\qquad
\lambda_2=-26,\quad\mathbf e_2=\frac{(-2,1,0)^T}{\sqrt5};\qquad
\lambda_3=-6,\quad\mathbf e_3=(0,0,1)^T.}
$$
With comparable prior uncertainty in these dimensionless coordinates and equally accurate scalar observations, prioritize the growing combination
$$
\boxed{y_{\rm obs}\propto x+2y.}
$$
An observation of this projection removes error in the direction that grows most strongly over a sufficiently short forecast; its <norm> initially grows at rate $2$ in Lorenz time units. The strongly contracting direction is a lower priority under those assumptions. This is not an observation of the absolute state amplitude alone: the relevant error projection is $(\delta x+2\delta y)/\sqrt5$. Anisotropic <Covariance matrix>[prior covariance], observation noise or a different forecast <norm> would modify the optimal choice; for a finite forecast the leading <right singular vector> of the actual <Fundamental matrix>[propagator] replaces this instantaneous direction.
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