My question is whehter or not I'm using the direct sum $\oplus$ and direct product $\otimes$ symbols correctly, or if I need a tensor product. I have no formal training with these symbols and if I am wrong below, I hope you will correct me.
The Cartesian plane is $\mathbb{R}^2=\mathbb{R}\oplus\mathbb{R}$ where $\oplus$ is the direct sum. For comlex numbers $\mathbb{C}$ and quaternions $\mathbb{H}$, it follows that $$ \mathbb{C}=\mathbb{R}\oplus i\mathbb{R}\qquad\text{and}\qquad\mathbb{H}=\mathbb{R}\oplus \mathbf{i}\mathbb{R}\oplus \mathbf{j}\mathbb{R}\oplus \mathbf{k}\mathbb{R}~~.$$
I want to take a product of $\mathbb{C}$ and $\mathbb{H}$ such that $$ \mathbb{C}\star\mathbb{H}=\mathbb{R}\oplus \mathbf{i}\mathbb{R}\oplus \mathbf{j}\mathbb{R}\oplus \mathbf{k}\mathbb{R}\oplus i\mathbb{R}~~,$$
where $\star$ is the mystery operation I'm asking about. If $\star$ is the operation I'm looking for, then a number in $\mathbb{C}\star\mathbb{H}$ is a 5-tuple: $$x\in\mathbb{C}\star\mathbb{H}\qquad\implies\qquad x=\big( x_1,x_2,x_3,x_4,x_5 \big).$$
Please tell me what I'm looking for or use a well defined counter example to explain why my question is ill-defined. Thanks!