Almost complex structure which fails to be compatible

Question

Let $V$ be a real vector space equipped with a scalar product $\langle, \rangle$ (i.e. a positive definite symmetric bilinear form).

We say that an endomorphism $J: V \to V$ is an almost complex structure if $J^2=-Id.$

$J$ is said to be compatible with the scalar product if $\langle J v, J w \rangle = \langle v, w \rangle. $

I'd like a very simple example of a scalar product and almost complex structure such that $J$ FAILS to be compatible with $\langle, \rangle.$ This is very basic -and hopefully trivial- but I can't find any counterexamples.

Answer 1

Take $\mathbb R^2$ with standard scalar product and $J(x,y)=(2y,-\frac12x)$.