How to show that in a self-dual code all words have weight divisible by 4 or half do it and the anther half don't. The title is basically what i have to do: Let $C$ be a binary self-dual code: Show that all words have weight divisible by four or half are divisible and the another half don't. I showed that basically whole words in $C$ have weight even. Also I tried defining a set $C_1=${$x\in C / w(x)$ is divisible by 4} and tried to create some function to show that $dim(C_1)\geq dim(C)-1$ but not succeeded.

Consider that there is a codeword $c$ with $4 \
mid w(c)$. Then look at the coset $c + C_{1}$. Can you show that every codeword in $C$ is either in $C_{1}$ or in $c+C_{1}$?

This implies that if not all codewords have weight 4, then the code consisting of weight 4 words of $C$ is an index 2 subgroup of $C$ (and so contains half the words).