Jacobson (Lie algebras, p.187) defines what is meant by a restricted Lie algebra: Def4: A restricted Lie algebra, $L$, of characteristic $p\not = 0$ is a Lie algebra of characteristic $p$ in which there is defined a mapping $a\rightarrow a^{[p]}$ such that
- $(\alpha a)^{[p]}=\alpha^p a^{[p]}$
- $(a+b)^{[p]}=a^{[p]}+b^{[p]}+\sum_{i=1}^{p-1}s_{i}(a,b)$, where $is_{i}(a,b)$ is the coefficient of $\lambda^{i-1}$ in $a(ad(\lambda a+b))^{p-1}$ and
- [$ab^{[p]}$]$=a(ad b)^p$
My questions lie in the understanding of how identity 2 comes about. On page 187 Jacobson introduces the polynomial ring $\mathfrak U(\lambda)$ and writes $(\lambda a+b)^p= \lambda ^{p} a^p+b^p +\sum_{i=1}^{p-1}s_{i}(a,b)\lambda^{i}$ where $s_{i}(a,b)$ is a polynomial in $a,b$ of total degree $p$. So in particular, for $p=2$ we have $(\lambda a+ b)^2=\lambda^2a^2+b^2+\lambda(ab+ba)$ so I conclude that $s_{1}(a,b)=ab+ba$
I have a two questions. First, at the bottom of page 187, he says, that if p=2, then $s_{1}(a,b)=$[$a,b$], but [$a,b$]$=ab-ba$, not $ab+ba$ like I got above. Where is my mistake?
Second, he differentiates $(\lambda a+b)^p= \lambda ^{p} a^p+b^p +\sum_{i=1}^{p-1}s_{i}(a,b)\lambda^{i}$ with respect to $\lambda$ and gets $\sum^{p-1}_{i=0}(\lambda a+b)^ia(\lambda a+b)^{p-i-1}=\sum_{i=1}^{p-1}is_{i}(a,b)\lambda^{i-1}$. I do not understand how the left hand side came about. I would like to see, or at least have it explained to me how to differentiate the left hand side. Thank you for your time.