This is Theorem 10.1.1 of Lind & Marcus's book, [An Introduction to Symbolic Dynamics and Coding](https://doi.org/10.1017/CBO9780511626302). They say that is "straightfordward" to go from



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> Let $X$ a shift of finite type and $Y$ a mixing shift of finite type such that $\text{Per}(X)\hookrightarrow\text{Per}(Y)$ and $h(X)<h(Y)$. Then, $X\hookrightarrow Y$.
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to



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> Let $X$ and $Y$ irreducible shift of finite type such that $\text{Per}(X)\hookrightarrow\text{Per}(Y)$ and $h(X)<h(Y)$. Then, $X\hookrightarrow Y$.
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How we can drop the mixing hypothesis on $Y$? I have thought in this all new year!



On Krieger's Embedding Theorem
the difference between the two statements is rather subtle. Of course proving the result with $X$ any SFT is more general than assuming $X$ to be irreducible, so there is nothing to do there. For $Y$, going from irreducible to mixing seems to be a stronger condition, however the structure threoy of (one-dimensional) SFTs tells us that a non-mixing irreducible SFT is merely a finite union of mixing ones, all conjugate to each other. If you have not seen this, take a look at Chapter 4.5 of Lind-Marcus.
Hence the conditions of the embedding theorem still hold and we might look at one of the cyclic components (i.e. a mixing SFT) of $Y$ instead.


