How to show that $\mathbb{R} [i]$ is not contained in the center of $\mathbb{H}?$

Question

Prove that $ \{a + bi | a, b \in \mathbb{R}\}$ is a subring of $\mathbb{H}$ which is a field but is not contained in the center of $\mathbb{H}$ where $\mathbb{H}$ denote Hamilton Quaternions

My attempt : A non-empty subset S of R is a subring if $a, b \in S ⇒ a - b,ab \in S$

take $R=\mathbb{H}$, $S=\mathbb{R} [i]=\{a + bi | a, b \in \mathbb{R}\}$

$a= a_1 + b_1 i , b= a_2 +b_2 i $

now $a-b = (a_1 -a_2) + (b_1 -b_2)i\in S \tag 1$

$ab= a_1a_2-b_1b_2 + (a_1b_2 + a_2b_1)i \in S \tag2$

From $(1)$ and $(2)$ we can say that $S=\mathbb{R} [i]=\{a + bi | a, b \in \mathbb{R}\}$ is subring of $\mathbb{H}$

My doubt :How to show that $\mathbb{R} [i]$ is not contained in the center of $\mathbb{H}?$

Answer 1

Your proof of the fact that it is a subring is fine.

And it is not contained in the center of $\Bbb H$ because $i\in\Bbb R[i]$ and $ij=-ji\ne ji$.