Isometries of real space of self-adjoint operators in Banach symmetric ideal

Let $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ be a separable or a perfect Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space $\mathcal H$ such that $\mathcal C_E \neq \mathcal C_2$, and let $\mathcal C_E^h=\{x\in \mathcal C_E: x=x^*\}$ be the real Banach subspace of self-adjoint operators in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$. A linear transformation $V$ on $\mathcal C_E^h$ is an isometry of $C_E^h$ onto itself if and only if there exists unitary or anti-unitary operator $u$ on $\mathcal H$ such that either $V(x)=uxu^*$ or $V(x)=-uxu^*$ for all $x \in \mathcal C_E^h$.