Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.

we know that (in this paper archive):

Theorem :The equation $\text{NPE}_d$ has integer solutions if and only if there exist two integers $a(d)=a$ and $b(d)=b$ such that $d=a^2+b^2$ and there exists a Pythagorean triplet $(A,B,C)$ such that $|aA-bB|=1$ and in this case $(Aa+Bb,C)$ is a solution.

Obviously if $\text{NPE}_d$ has integer solutions then $d$ cannot be divisible by any prime $p$ such that $p=3\mod 4$.

My

question: Is there any characterization for the integers $d$ for which $\text{NPE}_d$ and $\text{NPE}_{2d}$ have both integer solutions.

I used the characterization above, but I can't link the couple $(a(d),b(d))$ to $(a(2d),b(2d))$ because the theorem doesn't give us much information

The sequence of the elements $d$ for which $\text{NPE}_d$ is soluble is OEIS A031396.

I posted this question in math.exchange and does not receive any answer