Let $A$ be a central simple algebra over a field $F$. How one can see that the group scheme ${\bf\rm{PGL}}_1(A)$ is embedded as an open subgroup of $\mathbb{P}(A)$, the projective space over $A$.
Here ${\bf\rm{PGL}}_1(A)$ is given by the quotient ${\bf\rm{GL}}_1(A)/\mathbb{G}_m$ and the ${\bf\rm{GL}}_1(A)$ is a group scheme given by mapping any commutative, unital $F$-algebra $R$ to $(A_R)^*:=(A\otimes_FR)^*$.
Thanks!