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 $Q$.