Question: Show that $\mathfrak{sl}_3(\mathbb{C})$ and $\mathfrak{s}_3(\mathbb{C})$ are ideals in $\mathfrak{gl}_3(\mathbb{C})$ and $\mathfrak{gl}_3(\mathbb{C}) = \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{s}_3(\mathbb{C})$.
My attempt: For all $A \in \mathfrak{gl}_3(\mathbb{C})$. If $\text{tr}(A) = 0$ then $A \in \mathfrak{sl}_3(\mathbb{C})$; else $A = \frac{\text{tr}(A)}{3} I + (A - \frac{\text{tr}(A)}{3} I)$ with $\text{tr}(A - \frac{\text{tr}(A)}{3} I) = 0$, i.e., $A - \frac{\text{tr}(A)}{3} I \in \mathfrak{sl}_3(\mathbb{C})$. Clearly $\mathfrak{sl}_3(\mathbb{C}) \cap \mathfrak{s}_3(\mathbb{C}) = \{0\}$. Hence $\mathfrak{gl}_3(\mathbb{C}) = \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{s}_3(\mathbb{C})$.
Is my proof true and how to prove that $\mathfrak{sl}_3(\mathbb{C})$ and $\mathfrak{s}_3(\mathbb{C})$ are ideals? Thank all!