Property on Kronecker product

I read a paper and there was an equation which was finally derived an equivalent expression as

$$L = L_{T} \otimes I_{G} + I_{T} \otimes L_{G} = {\color{blue}{L_{T} \times L_{G}}} ,$$

and considering $$L_{T} = U_{T}\Lambda_{T} U_{T}^{*}$$ and $$L_{G} = U_{G}\Lambda_{G} U_{G}^{*}$$, it is obtained

$$L = {\color{red}{(U_{T}\otimes U_{G})(\Lambda_{T} \times \Lambda_{G})(U_{T} \otimes U_{G})^*}} ,$$

where $$\otimes$$ accounts for the Kronecker product, $$*$$ is the Hermitian and $$\times$$ is the Cartesian product.

I am confused how to obtain the results in blue and red. Would you please help me to clarify them?