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it is known quaternions are isomorphic to [tex]\mathcal{C}\ell^{+}_{3,0}[/tex], which is theeven subalgebraof [tex]\mathcal{C}\ell_{3,0}[/tex]

Is it possible to find an isomorphism between [tex]\mathcal{C}\ell_{2,0}[/tex] and [tex]\mathbb{H} \cong \mathcal{C}\ell^{+}_{3,0}[/tex] ?

*** my attempt was: ***

Let's consider [tex]\{1,e_1, e_2, e_{12}\}[/tex] and the morphismfdefined as follows:

[tex]f(1)=1[/tex]

[tex]f(e_{32})=e_1[/tex]

[tex]f(e_{13})=e_2[/tex]

[tex]f(e_{21})=e_{21}[/tex]

This almost works, in fact:

[itex]f(xy)=f(x)f(y)[/itex] always holds with one exception:

[tex]f(e_1 e_2) = -e_{21} = f(e_2)f(e_1)[/tex]

So, in this case, the morphism f does not preserve well the geometric product.

Is it possible to make the whole thing work?

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# Quaternions in Clifford algebras

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