Let $Q(i)$ be the extension of the rational numbers $Q$ obtained by adjoining a root i of the polynomial $X^2 + 1$.
Consider the algebra B defined by the Hilbert symbol $(-2, -5)$ over $Q(i)$. So, by definition,
$$
B := Q(i)[[\alpha, \beta]]/(\alpha^2 = -2, \beta^2 = -5, \alpha\beta = -\beta\alpha)
$$ 
here $[[\alpha, \beta]]$ means non-commutative polynomials in $\alpha, \beta$. The algebra $B$ is ramified at the primes of $Q(i)$ lying above $5$, and unramified at all other places.


My question: Does there exist an injection of $Q$-algebras
$$
B \to M\_4(Q), 
$$
where $M\_4(Q)$ is the ring of $4 \times 4$ matrices over $Q$. 



Morphism of algebras
No. Consider the matrix corresponding to $i$. It is semisimple with eigenvalues $i,i,-i,-i$ . So it's centralizer is $8$-dimensional - clearly, it is $M\_2(\mathbb Q(i))$. Since this algebra is ramified, it does not inject into the split algebra.


