Does anyone know how to prove that the following special value of the Modular Lambda Function is correct?

$$\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$$

I have a somewhat promising observation that might help us derive this special value, but it hasn't panned out for me so far. If we consider the elliptic function $\wp$ with $\tau=\sqrt{2}i$, the fundamental domains look like rectangles with side lengths of $1$ and $\sqrt{2}$, which exhibit a certain self-similarity: one of these rectangles can be dissected into two similar copies of the same rectangle, scaled down by a factor of $\sqrt{2}$. However, I haven't figured out how to use this fact to my advantage.

Another note: I did figure out how to prove the special value $\lambda(i)=1/2$: it follows trivially from the functional equation $\lambda(-1/\tau) =1-\lambda(\tau)$.

Any help is appreciated!