Let $H$ denote the standard field of quaternions over $\mathbb{Q}$. I know that it is a division ring (every non zero element has a inverse). It is a non-commutative division ring.
Let's denote its multiplicative group by $H^{*}$.
I have read that it is a $4$ dimensional $\mathbb{Q}$ - algebraic group. I am not able to write a proof.
Technically $H^*$ is not an algebraic group over $\mathbb Q$, but rather the set of rational points of an algebraic group $\mathbf H$ over $\mathbb Q$. (An algebraic group is a type of functor from fields to groups.)
Modulo this technicality, one way is to realize $H$ in $M_4(\mathbb Q)$ (pick a basis, and associate each element of $H$ to the 4x4 matrix given by this action). Then you can check $H^*$ is given by polynomial equations, and that it gives an algebraic group similar to the proof that GL($n$) is an algebraic group.