  In this note, we consider irreducible semigroups of real, complex, and
quaternionic matrices with real spectra. We prove Burnside type theorems in the
settings of reals and quaternions. First, we prove that an irreducible
semigroup of triangularizable matrices in $M_n(\mathbb{R})$ contains a vector
space basis for $M_n(\mathbb{R})$. In other words, $M_n(\mathbb{R})$ is the
only irreducible subalgebra of itself that is spanned by an irreducible
semigroup of triangularizable matrices in $M_n(\mathbb{R})$. Next, we use this
result to show that, up to similarity, $M_n(\mathbb{R})$ is the only
irreducible $\mathbb{R}$-algebra in $M_n(\mathbb{H})$ that is spanned by an
irreducible semigroup of matrices in $M_n(\mathbb{H})$ with real spectra. Some
consequences of our mains results are presented.
