Solution
= Solution
The antisymmetric contribution is the negative of twice the <cross-product matrix> for $\mathbf a$:
$$
M=\begin{pmatrix}1&2a_3&-2a_2\\-2a_3&1&2a_1\\2a_2&-2a_1&1\end{pmatrix}.
$$
The <cross-product matrix spectrum> is $0,\pm i|\mathbf a|$, so $M$ has eigenvalues $1,1\pm2i|\mathbf a|$. Multiplication gives
$$
\boxed{\det M=1+4|\mathbf a|^2.}
$$
Direct expansion of the displayed matrix yields the same real positive determinant.