Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 4 页 Section 1.3「Lie algebras of derivations」的代数、导子及内导子定义，第 9 页 Section 2「Ideals and Homomorphisms」Exercises 1 完整陈述（补充完整名词词形）
By an $F$-algebra (not necessarily associative) we simply mean a vector space $\mathfrak A$ over $F$ endowed with a bilinear operation $\mathfrak A\times\mathfrak A\to\mathfrak A$, usually denoted by juxtaposition (unless $\mathfrak A$ is a Lie algebra, in which case we always use the bracket). By a derivation of $\mathfrak A$ we mean a linear map $\delta:\mathfrak A\to\mathfrak A$ satisfying the familiar product rule $\delta(ab)=a\delta(b)+\delta(a)b$. It is easily checked that the collection $\operatorname{Der}\mathfrak A$ of all derivations of $\mathfrak A$ is a vector subspace of $\operatorname{End}\mathfrak A$. […] Since a Lie algebra $L$ is an $F$-algebra in the above sense, $\operatorname{Der}L$ is defined. Certain derivations arise quite naturally, as follows. If $x\in L$, $y\mapsto[xy]$ is an endomorphism of $L$, which we denote $\operatorname{ad}x$. In fact, $\operatorname{ad}x\in\operatorname{Der}L$, because we can rewrite the Jacobi identity (using (L2')) in the form: $[x[yz]]=[[xy]z]+[y[xz]]$. Derivations of this form are called inner, all others outer. […] 1. Prove that the set of all inner derivations $\operatorname{ad}x$, $x\in L$, is an ideal of $\operatorname{Der}L$.