Hermiticity and CAR of the ordered Ising Majoranas

The string $S_{j-1}=Z_1\cdots Z_{j-1}$ is Hermitian, has square $I$, and commutes with $X_j$ and $Y_j$ because it acts on different tensor factors. Hence ${\gamma }_{2j-1}$ and ${\gamma }_{2j}$ are Hermitian, and each squares to $I$.

On one site, ${\gamma }_{2j-1}{\gamma }_{2j}=S_{j-1}^2{X}_j{Y}_j=i{Z}_j$ and ${\gamma }_{2j}{\gamma }_{2j-1}=S_{j-1}^2{Y}_j{X}_j=-i{Z}_j$. Their sum is zero, so the two distinct operators at the same site anticommute.

Let $j<\mathrm{\ell }$ and choose either $X_j$ or $Y_j$ at site $j$ and either $X_{\mathrm{\ell }}$ or $Y_{\mathrm{\ell }}$ at site $\mathrm{\ell }$. The string in the later operator contains exactly one $Z_j$; all other factors commute across distinct sites, while both $X_j$ and $Y_j$ anticommute with $Z_j$ by [F2]. Moving the earlier site matrix past the later string therefore contributes exactly one minus sign. Thus every pair of operators at distinct sites anticommutes. Together with the squares from step 1.1 and the same-site computation in step 2.1, this gives the stated CAR for all indices.