$E\subset\mathbb{R}^n$ is an ellipsoid if $E = E(g):= \{x\in \mathbb{R}^n \mid x^t g x \le 1\}$ for some inner product $g$ on $\mathbb{R}^n$. Given an ellipsoid $E\subset\mathbb{R}^n$, how unique is $g$ such that $E=E(g)$? Is there a formula for $g$ such that $E=E(g)$ (see the note below for what kind of formula I envisage)? If $T\in SL(n,\mathbb{R})$ satisfies $E_1 = T(E_2)$ for two ellipsoids $E_1$ and $E_2$ of the same volume, does it follow that $T^t g_1 T = g_2$? What if $T$ is orthogonal or symplectic? Thanks for giving a hint or a reference.

Note: Given an ellipsoid $E$, then, I guess, $A_{ij} = \int_E (r^2 \delta_{ij} - r_i r_j) dV$ and $B_{ij} = \int_E r_i r_j d V$ are inner products; they don't give $g$, though, but are related to it. It is clear from these formulas that if $E_1 = T(E_2)$, then $T^t A_1 T = A_2$ provided $T\in SO(n)$, and $T^t B_1 T = B_2$ provided $T\in SL(n)$.