Why the real space of Hermitian matrices is composed of space of traceless Hermitian matrices direct sum $\text{Span}_{\mathbb{R}}(\{\mathbb{I}_X\})$

Question

I'm trying to understand a so-called fact (From the SM of article, proof of Lemma-SM 3) that $$\mathcal{h}(X)=\mathcal{h}_0(X)\oplus \text{Span}_{\mathbb{R}}(\{\mathbb{I}_X\}),$$ where $h(X)$ means the real space of all Hermitian matrices on system $X$ and $\mathcal{h}_0(X)$ means the space of all traceless Hermitian matrices on Hilbert space $X$. It seems like the right side is the full space of all Hermitian matrices, but I do not find any evidence to support this point.