# Order in quaternion algebra for Fuchsian group

It may be that I am missing something very simple. In S. Katok's book "Fuchsian Groups", Lemma 5.3.3, we have the following.

Lemma: Let $$\Gamma$$ be a Fuchsian group of finite covolume, $$k_0=\mathbb{Q}(\mathrm{tr}(\Gamma))$$. Assume that $$[k_0:\mathbb{Q}]<\infty$$ and $$\mathrm{tr}(\Gamma)\subset \mathcal{O}_{k_0}$$ (the ring of integers of $$k_0$$). Then, $$\mathcal{O}_{k_0}[\Gamma]=\left\{\sum_i a_i \gamma_i\,:\, a_i\in\mathcal{O}_{k_0},\,\gamma_0\in\Gamma\right\}$$ is an order of the quaternion algebra $$k_0[\Gamma]=\left\{\sum_i a_i\gamma_i\,:\, a_i\in k_0,\,\gamma_0\in\Gamma\right\}.$$

In the proof, we have the following argument.

We may assume that $$\gamma_0=\begin{pmatrix}\lambda\\&\lambda^{-1}\end{pmatrix}$$, $$\lambda\neq 1$$ and $$\Gamma\subseteq \mathrm{PSL}(2,K_0)$$ where $$K_0=k_0(\lambda)$$. If $$\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathcal{O}_{k_0}[\Gamma]$$, then $$a+d$$ and $$\lambda a+\lambda^{-1}d$$ are in $$\mathcal{O}_{k_0}$$. Notice that $$\lambda$$ and $$\lambda^{-1}$$ are units in $$K_0$$ and $$\mathcal{O}_{k_0}$$ is a subring of the ring of integers of $$K_0$$, so $$a$$ and $$d$$ are in the fractional ideal $$\frac{1}{\lambda^2-1} \mathcal{O}_{k_0}$$ of $$K_0$$.

I don't understand why $$a$$ is in that fractional ideal. It seemed to me that $$(\lambda^2-1)a=\lambda(\lambda a+\lambda^{-1}d)-(a+d)\in (\lambda-1)\mathcal{O}_{k_0}-\mathcal{O}_{k_0}.$$ But I don't see why that should lie in $$\mathcal{O}_{k_0}$$. Also, why is that ideal fractional? Wouldn't that imply that $$(\lambda^2-1)\in \mathcal{O}_{k_0}$$?

On the author's website, there is a mention of this Lemma in the errata: http://www.personal.psu.edu/sxk37/errata.pdf but it just says: "The end of the proof of Lemma 5.3.3 was modified according to M.Katz’s suggestion."

Thanks for any clarification!