# Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?

Why is it the case, that for Lorentz invariant Lagrangian $$\mathcal{L}$$, after Wick rotation, the $$O(4)$$ invariance is established, thus manifesting itself as having Euclidean metric? Is that a consequence of requiring the four vector fields to transform as $$A_0^E = iA_0$$ and $$A_j^E=E_j$$ or a result of it? So which premise comes first?
As long as the Minkowski action is constructed from Lorentz-covariant tensors, then under Wick rotation [where the contravariant and covariant $$0$$-components of the tensors are Wick-rotated in opposite ways], the corresponding Euclidean action becomes constructed from the corresponding $$O(4)$$-covariant tensors, cf. e.g. this Phys.SE post.
Note in particular that the Minkowski metric tensor [with the signature convention $$(-,+,+,+)$$] is Wick rotated to the Euclidean metric tensor.
• How does one postulate that? I get that after the wick rotation one gets $dx_1^2+ dx_2^2 _\dots$ so it certainly looks like Euclidean metric. But how can one infer from this information alone that the Lagrangian itself must be also one that adapts euclidean symmetry ? The fields themselves are not uniquely defined, but is the Lagrangian by analytic continuation uniquely determined, so as to leave $O(4)$ invariance as the only possible outcome ? Commented Jul 26, 2022 at 19:04