Exercise 13, § 13

The algebra of invariant polynomial functions of $\mathfrak{g}$ is generated:

a) in case $A_l$, by the functions $X \rightarrow$ Tr($X^i$)$, 2\leq i\leq l+ 1$;

b) in case $B_l$, by the functions $X \rightarrow$ Tr($X^{2i}$)$, 1\leq i\leq l$;

c) in case $C_l$, by the functions $X \rightarrow$ Tr($X^{2i}$)$, 1\leq i\leq l$;

d) in case $D_l$, by the functions $X \rightarrow$ Tr($X^{2i}$)$, 1\leq i\leq l-1$, and by one of the two polynomial functions $\widetilde{f}$ such that $\widetilde{f}(X)^2= (-1)^l$det($X$).