Thank you for your help! I'll have a look and try to figure out what I can do: I'm eager to learn more. Sorry about reversing Dickson's statement. Indeed, it was stupid of me!

Dear Will, thank you again! I will need however some more information about the link between genera and spinor genera with representability (you consider the case when they don't coincide, but what happens if they do?) I'm sure you may provide me with a reference for further reading which will help understanding things and finishing the "impossibility" proof (I do need to write a proof to this fact and thus need to learn!) Although your help has already been great. Thank you!

Dear Will, thank you for this explanation. If I may, I would like to ask a more precise question: is there a reference to an example of a ternary quadratic form that represents all negative integers over Q, but avoids some positive integer (I need forms over Q mostly). If no reference exists, is it possible to come up with a concrete example? In my mind representing every negative integer (over Q, not integrally) is quite a strong property to be checked.

That sounds interesting, although the quadratic form you brought up is isotropic and thus universal. My concern is that "negative universality" implies may be "total universality", but I fail to find a concrete example showing the opposite.

@individ: I'm interested to know if a form $f = a x^2 + b y^2 + c z^2$ with $a, b > 0$ and $c < 0$ can be "negatively universal" but avoid instead some positive integer. In principal, I need ternary quadratic forms with rational coefficients and look at which numbers they represent over rationals, although it seems my question won't change much in this setting.

@draks if you have more questions that you think I can answer or want to discuss maths (which is always a pleasure), please drop me a line by email. Hopefully you don't object personal communication,