Let $D_{1}$ and $D_{2}$ be two irreducible representations of a Lie algebra, of dimensions $\operatorname{dim} D_{1}$ and $\operatorname{dim} D_{2}$, and with Dynkin indices $\lambda_{D_{1}}$ and $\lambda_{D_{2}}$, respectively.

(i) Prove that the Dynkin index of the direct-sum representation $D_{1} \oplus D_{2}$ is given by $$\begin{equation*}
\lambda_{D_{1} \oplus D_{2}}=\lambda_{D_{1}}+\lambda_{D_{2}} \tag{6.416}
\end{equation*}$$
(ii) Prove that the Dynkin index of the tensor-product representation $D_{1} \otimes D_{2}$ is given by $$\begin{equation*}
\lambda_{D_{1} \otimes D_{2}}=\left(\operatorname{dim} D_{1}\right) \lambda_{D_{2}}+\left(\operatorname{dim} D_{2}\right) \lambda_{D_{1}} . \tag{6.417}
\end{equation*}$$