$[h_i,{(ad\ f_i)^t\ f_j}]=-(t\alpha_i+\alpha_j)(h_i){(ad\ f_i)^t\ f_j}$ in a Kac-Moody algebra

Question

In Carter's Lie algebras of finite and affine type, when constructing the Weyl group in the setting of a Kac-Moody algebras, Carter uses an identity $[h_i,{(ad\ f_i)^t(f_j)}]=-(t\alpha_i+\alpha_j)(h_i){(ad\ f_i)^t(f_j)}$. I'm having trouble seeing why this is the case? I've tried showing it by induction, but it doesn't seem to get me anywhere (the basecase is just from the defintion).

Answer 1

It seems that induction on $t$ works. As you noted, the case $t=0$ follows from the definition. Suppose $[h_i, (ad f_i)^t (f_j)]=-(t\alpha_i + \alpha_j)(h_j)(ad f_i)^t(f_j) $ and write $x=(ad f_i)^t (f_j)$ for simplicity.

Then $(ad f_i)^{t+1}(f_j) = [f_i, x] $ and \begin{align} [h_i, (ad f_i)^{t+1}(f_j)] &= [f_i, [h_i, x]]-[x,[h_i, f_i]] &&\text{Jacobi identity} \\ &=[f_i, -(t\alpha_i+\alpha_j)(h_i)x]+[x,\alpha_i(h_i)f_i] &&\text{induction hypothesis}\\ &=-(t\alpha_i+\alpha_j)(h_i)[f_i,x]-\alpha_i(h_i)[f_i, x] \\ &=-\left( (t+1)\alpha_i + \alpha_j \right)(h_i)[f_i, x] \\ &= -\left( (t+1)\alpha_i + \alpha_j \right)(h_i)(ad f_i)^{t+1}(f_j) \end{align}

as desired.