= Solution
The association between four real coordinates and Hermitian two-by-two <matrices> is a real-linear bijection. Congruence by $S$ preserves Hermiticity, so $X'=SXS^\dagger$ defines another real <four-vector> depending linearly on the original. Its <determinant> is
$$
\det X=(ct)^2-x^2-y^2-z^2,\qquad
\det X'=\det S\,\det X\,\det S^\dagger=\det X.
$$
Thus the induced invertible linear map preserves the Minkowski quadratic form:
$$
\boxed{\Lambda^{\mathsf T}\eta\Lambda=\eta,\qquad
\eta=\operatorname{diag}(1,-1,-1,-1).}
$$
The maps obtained continuously from the identity are proper and orthochronous. For example, a future timelike <vector> gives a positive-definite $X$, which remains positive definite under invertible congruence; its <time> direction is preserved. The <matrices> $S$ and $-S$ induce the same <four-vector> transformation, illustrating the <Lorentz spinor double cover>.
The source uses $x+iy$ in the upper-right entry, corresponding to $X=ctI+x\sigma_x-y\sigma_y+z\sigma_z$ for the standard <Pauli matrices>. This is a permissible basis convention, but its minus sign in the $y$ basis must also be retained in the <spinor> identification.
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