N. J. Kalton wrote in his 1982 paper Convexity Conditions for Non-Locally Convex Lattices a statement that
lattice 1-convexity is equivalent to normability.
I cannot see why this is immediate. Kalton did not point to any reference of this statement neither.
Related notations are as follows.
$\|\cdot\|$ is called a quasi-norm on $X$ if $\|x\|>0$ for $x\in X, x\ne 0$, $\|\alpha x\|=|\alpha|\|x\|$ for $\alpha\in\mathbb{R}, x\in X$, $\|x_{1}+x_{2}\|\leq C(\|x_{1}\|+\|x_{2}\|)$ for $x_{1},x_{2}\in X$, some constant $C>0$.
$X$ is said to be normable if there is a norm $\|\cdot\|'$ such that $c\|\cdot\|\leq\|\cdot\|'\leq c\|\cdot\|$ for some constant $c>0$.
$X$ is said to be $1$-convex if \begin{align*} \left\|\sum_{i=1}^{n}|x_{i}|\right\|\leq M\sum_{i=1}^{n}\|x_{i}\| \end{align*} for $x_{1},...,x_{n}\in X$, some constant $M>0$. Here we assume that $X$ has an order lattice and so $|x_{i}|$ is an element of the lattice space $X$.
Just found a hint somewhere else.
Consider \begin{align*} \|x\|'=\inf\left\{\sum_{i=1}^{n}\|x_{i}\|: x=\sum_{i=1}^{n}x_{i}, x_{i}\in X\right\}, \end{align*} then $\|\cdot\|'$ is a norm satisfying that \begin{align*} M^{-1}\|x\|\leq\|x\|'\leq\|x\|. \end{align*}