Prove $G = \{ a + b \hat{i} +c \hat{j} +d \hat{k} : a,b,c,d \in \mathbb{R} \}$ is a locally compact topological group Let $G = \{ a + b \hat{i} +c \hat{j} +d \hat{k} : a,b,c,d \in \mathbb{R} \}$ be the group of non-null quaternions. I need to prove that $G$ is locally compact topological group by showing that $G$ can be viewed as an open subset of $\mathbb{R^4}$ .  
Also can someone please explain how to compute the Haar measure on $G$ ? 
 A: The quaternions are in bijection with $\mathbb{R}^4$ via $a+bi+cj+dk \mapsto (a,b,c,d)$.
The nonzero quaternions are then in bijection with $G = \mathbb{R}^4 \setminus \{(0,0,0,0)\}$.  An open subset of a locally compact topological space is locally compact.
Let's check that multiplication of quaternions is a continuous function.  By the universal property of the product topology, we just need to check that multiplcation $X \times X \rightarrow \mathbb{R}^4$ is a continuous function componentwise into $\mathbb{R}^4$.  For example, if you multiply $(a+bi+cj+dk)(a'+b'i+c'j+d'k)$ and collect the constant term, you get $aa' -bb' - cc' - dd'$, which is definitely continuous as a function of the variables $a,a',b,b'$ etc.  
Let's compute the Haar measure on $G$.  Define a positive linear functional $T: C_c(G) \rightarrow \mathbb{C}$ by
$$T(f) = \int_{-\infty}^{\infty}  \int_{-\infty}^{\infty}  \int_{-\infty}^{\infty}  \int_{-\infty}^{\infty} \frac{f(x,y,z,w)}{x^2 + y^2 + z^2 + w^2} dx \space dy \space dz \space dw$$
You have to check that $T$ is well defined on those functions $f: \mathbb{R}^4 \setminus \{(0,0,0,0) \}$ which are continuous and of compact support.  By the Riesz representation theorem, there will exist a unique Radon measure $\mu$ on the Borel sets of $G$ such that $T(f) = \int\limits_G f d\mu$ for all $f \in C_c(G)$.  You also have to check that if $g \in G$, and $f \in C_c(G)$, then $T(L_g(f)) = T(R_g(f)) = T(f)$, where $L_g(f)(x) = f(gx)$ and $R_g(f)(x) = f(xg)$.  This implies that $\mu$ is moreover a left and right Haar measure.
