For lie algebra, may I ask why $\operatorname{ad}_x[y,z]=[[x,y],[x,z]]$?
Tbe following set of notes claim this is from the Jacobi identity: https://www.math.hkust.edu.hk/~emarberg/teaching/2023/Math5143/lectures/06_Math5143_Spring2023.pdf
However, the Jacobi identity means we have $$ \operatorname{ad}_x[y, z]=\left[\operatorname{ad}_x y, z\right]+\left[y, \operatorname{ad}_x z\right] $$ but I don't think this is the same as what is given above in the picture.
The claimed identity $$ [x,[y,z]]=[[x,z],[y,z]] $$ is false. As an example, let $L=\mathfrak{sl}_2(K)$ with basis $(e, f,h)$ and $[e,f]=h$, $[h,e]=2e$ and $[h,f]]=-2f$. Assume it would hold. Then $$ 0=[h,h]=[h,[e,f]]=[[h,e],[h,f]]=[2e,-2f]=-4h. $$ which is a contradiction.
In the proof we do not need this "equality". We only need that $[L,[I,L]\subseteq [I,L]$, which just follows from the Jacobi identity.