3 problems

First-Casimir conjecture. For every choice of the real constants $a_k$, there exist real constants $b_k$ such that

Adjoint eigenvalue conjecture. For arbitrary even $p$,

Generation conjecture. Let $T(B)$ be the algebra generated by $T^k(a_1,\ldots,a_k)$ where $a_1,a_2,\ldots,a_k\in B$ and $k\in\mathbb{Z}_{+}$. Then