Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality $${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E.$$ This is how one can prove that every Euclidian space satisfies (H).

Now consider the Euclidian space ${\mathbb R}^n$.

Does $E={\mathcal L}({\mathbb R}^n)$, endowed with the operator norm, satisfy (H) ? If so, is it $\ell^1$-embeddable ?

Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality $${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E.$$ This is how one can prove that every Euclidian space satisfies (H).

Now consider the Euclidian space ${\mathbb R}^n$.

Does $E={\mathcal L}({\mathbb R}^n)$, endowed with the operator norm, satisfy (H) ? If so, is it $\ell^1$-embeddable ?

Edit. After Mateusz' answer, the two-dimensional case remains open.