Why there is no $5$-dimensional analogous of Quaternions?

Question

Why there is no $5$-dimensional analogous of Quaternions? Why the following definition is not well-defined? $$i^2=j^2=k^2=\ell^2=ijk=jk\ell=k\ell i=-1,\quad ijkl=1.$$

Answer 1

If you are looking for a real vector space $V$ with basis $\{1,i,j,k,\ell\}$ and an associative product such that $V$ becomes an $\Bbb{R}$-algebra, then we run into the following difficulty.

The rule that $i^2=-1$ means that $V$ also has a structure as a vector space over the field $\Bbb{C}=\Bbb{R}(i)$.

But a vector space over $\Bbb{C}$ necessarily has an even dimension as a vector space over $\Bbb{R}$.

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The above is probably not the shortest route to a contradiction (see the comment by verret). But it also rules out many modifications to the suggested relations defining the product.