On Grothendieck–Serre conjecture in mixed characteristic for $SL_{1,D}$

Let $R$ be an unramified regular local ring of mixed characteristic, $D$ an Azumaya $R$-algebra, $K$ the fraction field of $R$, $Nrd: D^{\times} \to R^{\times}$ the reduced norm homomorphism. Let $a \in R^{\times}$ be a unit. Suppose the equation $Nrd=a$ has a solution over $K$, then it has a solution over $R$. Particularly, we prove the following. Let $R$ be as above and $a,b,c$ be units in $R$. Consider the equation $T^2_1-aT^2_2-bT^2_3+abT^2_4=c$. If it has a solution over $K$, then it has a solution over $R$.