$\textbf{Definition}$: A code $\mathcal{C}$ is self-orthogonal if $\mathcal{C}\subseteq \mathcal{C}^{\perp}$.
I need proof that if $\mathcal{C}$ is binary code self-orthogonal. I want to proof that $\textbf{1}=11\cdots 1 \in \mathcal{C}^{\perp}$.
I already proved that if the hypotheses are fulfilled then each codeword in $\mathcal{C}$ have weight even but i don´t know proof that $\textbf{1}\in \mathcal{C}^{\perp}$.
A binary string is self-orthogonal iff it's orthogonal to $\textbf{1}$. (Both cases happen iff the number of $1$'s in it is even).
If a code is self-orthogonal, clearly any string in it is self-orthogonal, os $\textbf{1}$ is orthogonal to any string in the code.
Edit: in other terms, by definiteion $\textbf{1} \in \mathcal{C}^\perp $ iff for every $c\in\mathcal{C}$, $\textbf{1}\cdot c=0$. But $\textbf{1}\cdot c$ is the number of $1$'s in $c$ (mod $2$), and you already proved that if $c\in\mathcal{C}$ then that number is even. Hence, for every $c\in\mathcal{C}$, $\textbf{1}\cdot c=0$.