Clifford Algebra decomposition I am trying to show the three dimensional Clifford algebra over the reals $C_3 \cong \mathbb{H} \oplus \mathbb{H}$.
where $e_1,e_2,e_3$ are generators of $C_3$.  I found two idempotents $k_1 = (1 + e_1e_2e_3)/2$ and $k_2 = (1- e_1e_2 e_3)/2$ (orthogonal) so I am trying to use the idea that
$C_3 \cong C_3 k_1 \oplus C_3 k_2$.
I am struggling to show $C_3k_1 = \mathbb{H} = C_3k_2$. 
 A: I assume that you have a signature $\mathcal{C}l(0,3)$ but the method should work with the other signatures as well.
Consider the general  element
$A=
{{a}_{1}}+{{e}_{1}}\,{{a}_{2}}+{{e}_{2}}\,{{a}_{3}}+{{e}_{3}}\,{{a}_{4}}+{{a}_{5}}\,\left( {{e}_{1}} {{e}_{2}}\right) +{{a}_{6}}\,\left( {{e}_{1}} {{e}_{3}}\right) +{{a}_{7}}\,\left( {{e}_{2}} {{e}_{3}}\right) +{{a}_{8}}\,\left( {{e}_{1}} {{e}_{2}} {{e}_{3}}\right) $
The elements of the event sub-algebra are of the form
$A_2 = {{a}_{1}}+{{a}_{5}}\,\left( {{e}_{1}} {{e}_{2}}\right) +{{a}_{6}}\,\left( {{e}_{1}} {{e}_{3}}\right) +{{a}_{7}}\,\left( {{e}_{2}} {{e}_{3}}\right) $
The dual of an element of the odd sub-algebra is of the form
$-A_1^\star = i A_1 = {{a}_{8}}-{{a}_{4}}\,\left( {{e}_{1}} {{e}_{2}}\right) +{{a}_{3}}\,\left( {{e}_{1}} {{e}_{3}}\right) -{{a}_{2}}\,\left( {{e}_{2}} {{e}_{3}}\right) $,
where $i$ is the pseudo-scalar.
Finally, the bi-vector multiplication table is
$\begin{pmatrix}1 & {{e}_{1}} {{e}_{2}} & {{e}_{2}} {{e}_{3}} & {{e}_{1}} {{e}_{3}}\\
{{e}_{1}} {{e}_{2}} & -1 & -{{e}_{1}} {{e}_{3}} & {{e}_{2}} {{e}_{3}}\\
{{e}_{2}} {{e}_{3}} & {{e}_{1}} {{e}_{3}} & -1 & -{{e}_{1}} {{e}_{2}}\\
{{e}_{1}} {{e}_{3}} & -{{e}_{2}} {{e}_{3}} & {{e}_{1}} {{e}_{2}} & -1\end{pmatrix} $
Which is the quaternion multiplication table. 
So we can define a morphism
$ H: A \mapsto (A_2,  i A_1)  $
exhibiting the desired structure.
