I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$. Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real forms of ${\rm SL}(2,{\Bbb C})$ are of the form ${\rm SU}(2,F_\lambda)$, where $F_\lambda$ is the diagonal Hermitian form on ${\Bbb C}^2$ with matrix ${\rm diag}(1,\lambda)$, $\lambda$ taking the values 1 and $-1$.

Question. Is it true that any ${\Bbb Q}$-form of ${\rm SL}(2,{\Bbb C})$ is isomorphic to ${\rm SU}(2,F_{K,\lambda})$, where $F_{K,\lambda}$ is the diagonal Hermitian form on $K^2$ for some quadratic extension $K/{\Bbb Q}$ with matrix ${\rm diag}(1,\lambda)$, for some $\lambda\in {\Bbb Q}^\times$ ?