Let $F$ be a subfield of the real numbers field, $\mathbb{R}$. Let $S$ be an irreducible semigroup of triangularizable matrices in $M_{n}(F)$. If $S'$ be a semigroup of matrices in $M_{n}(\mathbb{R})$ such that $S=S'|_{F^{n}}$, $S$ is the restriction of $S'$ over $ F^{n} $ or $S'$ is the extention of $S$ over $ \mathbb{R}^{n}$. If $A$ is a one-rank matrix in $Alg_{\mathbb{R}}(S')$, the algebra spanned by $S'$ over $\mathbb{R}$, then there is a one-rank matrix in $Alg_{F}(S)$, the algebra spanned by $S$ over $F$.
Why? What is the proof? Is it correct that $A|_{F^{n}}$ is one-rank in $Alg_{F}(S)$?