Searching simple identities between power sums of roots?

Question

The Newton-Girard identities relating Vieta coefficients $e_k$ with power sums $P_k$ of roots of a polynomial, namely,$$ke_k = \sum_{i = 1}^k (-1)^{i - 1}e_{k-i}P_k$$ {with $e_1$ = $P_1$, and $e_k$ = 0 for $k$ exceeding the degree of polynomial in consideration} enables after some careful rearranging, an "inverse expression" for a power sum $P_k$ in terms of lower values of $k$. For instance, the familiar $2e_2$ = ${P_1}^2 - P_2$ ; $6e_3$ = ${P_1}^3 - 3P_1P_2 + 2P_3$, and then on derived expressions start getting rapidly complex: $$24e_4 = {P_1}^4 - 6{P_1}^2P_2 + 3{P_2}^2 + 8P_1P_3 - 6P_4.$$ Thus the expression I derive from the Newton-Girard identity $5e_5 = e_4P_1 - e_3P_2 + e_2P_3 - e_1P_4 + P_5$ turns into the pretty messy $$120e_5 = {P_1}^5 - 10{P_1}^3P_2 + 15P_1{P_2}^2 + 20{P_1}^2P_3 - 30P_1P_4 - 20P_2P_3 + 24P_5.$$ Lately, however, I stumbled on a thread where someone not only gets rid of the $e_5$ but provides an amazingly simpler and neat order-5 expression for a cubic :$$6P_5 = {P_1}^5 - 5({P_1}^3P_2 - {P_1}^2P_3 - P_2P_3)$$ identity obtained, he says by a method involving bruteforce using some matrix methods and Cramer's rule, an explanation about reduction of a system of equations which I couldn't grasp. I was wondering if such simple identities exist or can be derived for orders 6, 7 or even higher for a cubic polynomial ; I've never encountered any so far in textbooks or in mathematical forum discussions !