Compute the reduced norm of an invertible fractional $\mathcal{O}$-ideal in a quaternion algebra

Question

Let $B$ be a Quaternion algebra over $\mathbb{Q}$. Let $\mathcal{O}$ be a maximal order in $B$ and $I$ be an invertible (left) fractional $\mathcal{O}$-ideal. Suppose $I=\mathcal{O}\alpha_1+\cdots+\mathcal{O}\alpha_n$ for some $\alpha_i \in B$. How can we compute the reduced norm of $I$? I know invertibility means $\text{nrd}(I)^2=[O:I]$ in quaternion algebras and maybe we can compute $[O:I]$ from the generators $\alpha_i$?