galois norm and trace of field extensions

Question

Let $K \subset L \subset E$ and let Nm$_{E/K}(x)$ and Tr$_{E/K}(x)$ be its norm and trace, the determinant and trace of $x$ acting by multiplication on $E$. How can one show that $$ Nm_{E/K}(x)=Nm_{L/K}(Nm_{E/L}(x)), Tr_{E/K}(x)=Tr_{L/K}(Tr_{E/L}(x))? $$ It's easy for separable extensions, but I have no idea for arbitrary one.