Quadratic Forms in $k^n$ for a general field $k$

Question

I've read here that any quadratic form over $k^n$ with $n\in\mathbb{N}$ and $\text{char}(k)\neq 2$ is diagonalizable.

But considering $k=\mathbb{Q}$, which has characteristic $0$, the quadratic form $2x^2+2xy+y^2$ defined in $k^2$ is not diagonalizable, right?

What am I missing?