 Solution

The chain alternates at every step between the two classes $\{1,2\}$ and $\{3,4\}$. It is irreducible and has period two: returns require an even number of steps, and a two-step return has positive probability.

Writing the off-diagonal block as $B=\left(\begin{array}{cc}1/3& 2/3\\ 2/3& 1/3\end{array}\right)$, its square is $B^2=\left(\begin{array}{cc}5/9& 4/9\\ 4/9& 5/9\end{array}\right)$. HenceEvery two steps remain within the initial class, where the sampled chain is irreducible and aperiodic.