= Solution
First divide each of the first two equations by two. Then
$$
\boxed{K=\begin{pmatrix}3/2&-1/2&0\\-1/2&3/2&0\\0&0&4\end{pmatrix}}.
$$
The equal-component and opposite-component directions diagonalize the upper block, while the third coordinate is independent. Direct multiplication gives the <eigenvalues> and unit <eigenvectors>
$$
\boxed{\begin{array}{c|c}
\gamma&\text{normalized eigenvector}\\\hline
1&\frac1{\sqrt2}(1,1,0)^T\\
2&\frac1{\sqrt2}(1,-1,0)^T\\
4&(0,0,1)^T
\end{array}}.
$$
They form an <orthonormal eigenbasis> because $K$ is a real <symmetric matrix>.
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