Currently I'm reading the paper by Pluta and Russo titled Ternary operator categories.
According to section $1.2$ of the paper, an associative triple system is a vector space $V$ with a trilinear map $(x,y,z) \to [x,y,z]$ of $V^3$ into $V$ and satisfies $[[x,y,z],u,v]=[x,y,[z,u,v]]= [x,[u,z,y],v]$. For a given normed associative triple system $M$, define $$E(M)=\operatorname{End}(M)\oplus\overline{End(M)}^{op}$$ Where the notation $\overline{V}$ for a complex vector space means that the scalar multiplication in $V$ is $(\lambda,v)=\overline{\lambda}v$, End($M$) denotes the space of endomorphisms of $M$ and $\overline{End(M)}^{op}$ denotes the opposite of $\overline{End(M)}$
Can someone clarify me the norm in $E(M)$
The paper is not clear on that, and I have no access to the references. I would tend to assume that the natural way to do this is to use the operator norm on $\operatorname{End}(M)$, namely $$ \|\phi\|=\sup\{\|\phi(x)\|:\ x\in V,\ \|x\|=1\}, $$ and $$ \|\phi\oplus\psi\|=\max\{\|\phi\|,\|\psi\|\}. $$