I suspect that if the standard form of a code is $$H = \begin{pmatrix} H_X & 0 \\ 0 & H_Z \end{pmatrix}~, \quad(1)$$ then I can claim that the code is CSS.
They way I'm thinking about this problem is a scenario in which someone hands me a check matrix $H'$, they don't tell me whether it's CSS or not, and it's not in the form (1). Can I claim that it's CSS if its standard form is like (1)?
In CSS Code in disguise it is shown that a stabilizer code $H$ is a CSS code if and only if $H$ has transversal CNOT. In principle I could use this to check whether $H'$ is CSS or not, but it is a procedure whose cost scales exponentially with the size of $H'$ and I would like to find a cheaper alternative.
