Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 of his paper "Heights and the Special Values of L-series", Gross constructs a genus 0 curve $Y$ over $\mathbb{Q}$ associated to the quaternion algebra $B$ as follows.

For any $\mathbb{Q}$-algebra $E$, the points of $Y$ in $E$ are given by

$$ Y(E) = \{ \alpha \in B \otimes_{\mathbb{Q}} E \mid \operatorname{Tr}{(\alpha)} = 0 = \operatorname{N}(\alpha) \} / E^{\times} $$

where $\operatorname{Tr}(\cdot)$ and $\operatorname{N}(\cdot)$ are the reduced trace and the reduced norm in the quaternion algebra $B$, given as follows. For $h = x + iy + jz + ijw \in B$, with $x, y, z, w \in \mathbb{Q}$,

\begin{align} \operatorname{Tr}(h) &= h + \overline{h} = 2x\\ \operatorname{N}(h) &= h\overline{h} = x^2 - ay^2 - bz^2 + abw^2 \end{align}

Now, my confusion is that then I'm not sure about what it means to take the reduced trace and norm of an element $\alpha \in B \otimes_{\mathbb{Q}} E$.

Finally, I have also seen in another paper describing the same construction, that then combining the two equations $\operatorname{Tr}{(\alpha)} = 0 = \operatorname{N}(\alpha)$, $Y$ is basically the conic $ay^2 + bz^2 = abw^2$.

**Questions**

How should I interpret taking the trace $\operatorname{Tr}{(\alpha)}$ and norm $\operatorname{N}{(\alpha)}$ for $\alpha \in B \otimes_{\mathbb{Q}} E$ ?

According to the last observation about $Y$ being the conic $ay^2 + bz^2 = abw^2$, does that mean that $Y(E)$ can be thought of as $\{ (y, z, w) \in E^3 \mid ay^2 + bz^2 = abw^2 \}$?

Thank you very much for any help.