# Nilpotent and solvable Lie algebras: Engel's and Lie's theorems *Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).* An algebra of upper-triangular operators preserves a flag. An algebra of strictly upper-triangular operators does more: every sufficiently long product of its operators is zero. Engel's and Lie's theorems recover these matrix pictures from intrinsic conditions. Their conclusions look similar, but their hypotheses and their dependence on the ground field differ. The prerequisite is [Lie algebras: definitions, examples and first constructions](RT-LIE-01.md). We use its ideals, quotients, adjoint representation and module operations. All Lie algebras and representation spaces in this lesson are finite-dimensional unless a paragraph explicitly says otherwise. Engel's theorem works over every field. Lie's theorem will require an algebraically closed field of characteristic zero. The basic reference is [Milne, Lie algebras]. ## 1. Two ways brackets can run out For a Lie algebra \(\mathfrak g\), define the lower central series and the derived series by \[ C^1\mathfrak g=\mathfrak g,\qquad C^{r+1}\mathfrak g=[\mathfrak g,C^r\mathfrak g],\qquad D^0\mathfrak g=\mathfrak g,\qquad D^{r+1}\mathfrak g=[D^r\mathfrak g,D^r\mathfrak g]. \] Each term is an ideal. For the lower central series this follows inductively from Jacobi: \([x,[y,z]]=[[x,y],z]+[y,[x,z]]\). For the derived series the same identity applies with both bracket arguments in the preceding ideal. The series are decreasing. The algebra is **nilpotent** if \(C^{c+1}\mathfrak g=0\) for some \(c\), and **solvable** if \(D^r\mathfrak g=0\) for some \(r\). The zero algebra satisfies both definitions. The inclusion \(D^r\mathfrak g\subseteq C^{r+1}\mathfrak g\), proved by induction using the decreasing central series, shows that nilpotence implies solvability. The reverse implication fails. For example, in the algebra with basis \(x,y\) and \([x,y]=y\), \[ D^1\mathfrak g=ky,\qquad D^2\mathfrak g=0,\qquad C^r\mathfrak g=ky\quad(r\ge2). \] It is solvable but not nilpotent. Its centre is zero. In contrast, the Heisenberg algebra has \(C^2=kz\) and \(C^3=0\). There is a useful way to count matrix brackets. Let \(F^r\) be the space of \(n\times n\) matrices whose possibly nonzero entries satisfy \(j-i\ge r\). Then \[ F^rF^s\subseteq F^{r+s},\qquad [F^r,F^s]\subseteq F^{r+s}. \] Indeed, a nonzero summand \(X_{ik}Y_{kj}\) requires \(k-i\ge r\) and \(j-k\ge s\). The strictly upper-triangular algebra is \(F^1\), so \(C^r\mathfrak n_n\subseteq F^r\) and \(C^n\mathfrak n_n=0\). Its nilpotence class is \(n-1\) when \(n\ge2\): successive brackets of \(E_{12},E_{23},\ldots,E_{n-1,n}\) produce \(E_{1n}\ne0\). The upper-triangular algebra \(\mathfrak b_n\) satisfies \([\mathfrak b_n,\mathfrak b_n]\subseteq F^1\), because a commutator has zero diagonal. Consequently \(D^r\mathfrak b_n\subseteq F^{2^{r-1}}\) for \(r\ge1\); it is solvable. For \(n\ge2\) it is not nilpotent, since it contains \(E_{11},E_{12}\) with \([E_{11},E_{12}]=E_{12}\). For \(\mathfrak b_2\), this is an exact calculation: \(D^1=kE_{12}\), \(D^2=0\), and \(C^r=kE_{12}\) for every \(r\ge2\). The first equality follows from the zero diagonal of every commutator and the displayed bracket. Bracketing \(E_{12}\) again with \(E_{11}\) proves that the central series never terminates. **Proposition 1.1 (passing to smaller algebras).** Subalgebras and quotients of nilpotent algebras are nilpotent. Subalgebras and quotients of solvable algebras are solvable. If \(I\) is a solvable ideal and \(\mathfrak g/I\) is solvable, then \(\mathfrak g\) is solvable. **Proof.** The relevant series of a subalgebra lies in the series of the ambient algebra. A surjective homomorphism maps each series term onto the corresponding quotient term, by induction from bracket preservation. Finally, if \(D^a(\mathfrak g/I)=0\), then \(D^a\mathfrak g\subseteq I\). If \(D^bI=0\), repeated bracketing gives \(D^{a+b}\mathfrak g\subseteq D^bI=0\). \(\square\) An extension of nilpotent algebras need not be nilpotent: the preceding two-dimensional example has nilpotent ideal \(ky\) and nilpotent quotient \(kx\). **Proposition 1.2 (extending the field).** For a field extension \(K/k\), the algebra \(K\otimes_k\mathfrak g\) is nilpotent, respectively solvable, exactly when \(\mathfrak g\) is. **Proof.** Bilinearity gives \([K\otimes U,K\otimes W]=K\otimes[U,W]\). Induction identifies both series after extending the field. A nonzero vector space remains nonzero under field extension, as any basis shows, so termination before and after extension is equivalent. \(\square\) ## 2. Engel's theorem: a common zero vector An endomorphism \(X\) is nilpotent if \(X^r=0\) for some \(r\). This says something about an action, whereas nilpotence of a Lie algebra says something about its brackets. The adjoint action connects the two. We first need an elementary observation. On \(\operatorname{End}(V)\), write \(\operatorname{ad}X=L_X-R_X\), where \(L_X(T)=XT\) and \(R_X(T)=TX\). These two operators commute. If \(X^r=0\), then \[ (\operatorname{ad}X)^{2r-1} =\sum_{j=0}^{2r-1}(-1)^j \binom{2r-1}{j}L_X^{\,2r-1-j}R_X^j=0. \] Every summand has at least \(r\) copies of \(L_X\) or \(R_X\). This proof uses no division, so it is valid in every characteristic. Restrictions to invariant subspaces and induced maps on quotients are also nilpotent. **Theorem 2.1 (Engel, linear form).** Let \(V\ne0\) and let \(\mathfrak l\subseteq\mathfrak{gl}(V)\) be a Lie subalgebra all of whose elements are nilpotent endomorphisms. There is \(v\ne0\) such that \(Xv=0\) for every \(X\in\mathfrak l\). There is a basis in which all elements of \(\mathfrak l\) are strictly upper-triangular. **Proof.** Induct on \(\dim\mathfrak l\). The zero algebra is immediate. Assume the assertion about a common zero vector is known for smaller-dimensional operator algebras. For every proper subalgebra \(\mathfrak k\subsetneq\mathfrak l\), let \(\mathfrak k\) act on \(\mathfrak l/\mathfrak k\) by brackets. This action is well-defined because \([\mathfrak k,\mathfrak k]\subseteq\mathfrak k\). Every acting operator is nilpotent by the observation above. Its image has dimension at most \(\dim\mathfrak k<\dim\mathfrak l\), so induction supplies a nonzero class \(Z+\mathfrak k\) killed by \(\mathfrak k\). Thus \([\mathfrak k,Z]\subseteq\mathfrak k\), and the normalizer of \(\mathfrak k\) strictly contains \(\mathfrak k\). Choose a maximal proper subalgebra \(\mathfrak k\). Its normalizer is therefore \(\mathfrak l\), so it is an ideal. It has codimension one: if \(\dim(\mathfrak l/\mathfrak k)>1\), a one-dimensional subspace of this quotient is a proper nonzero subalgebra, contradicting maximality after taking its inverse image. Induction on \(\mathfrak k\subseteq\mathfrak{gl}(V)\) gives \[ W=\{v\in V:Kv=0\text{ for all }K\in\mathfrak k\}\ne0. \] This subspace is \(\mathfrak l\)-invariant. Indeed, \(K(Xw)=X(Kw)+[K,X]w=0\), since \(\mathfrak k\) is an ideal. Write \(\mathfrak l=\mathfrak k+kZ\). The nilpotent operator \(Z|_W\) has nonzero kernel: otherwise all its powers would be injective, contrary to a power being zero. A nonzero vector in that kernel is killed by all of \(\mathfrak l\). This completes the induction. The line \(kv\) is killed by \(\mathfrak l\). On \(V/kv\) the induced operators remain nilpotent, so repeat the common-vector assertion. We obtain a flag \[ 0=V_0\subset V_1\subset\cdots\subset V_d=V,\qquad \dim V_i=i,\qquad \mathfrak lV_i\subseteq V_{i-1}. \] A basis adapted to this flag gives the claimed matrices. \(\square\) **Corollary 2.2 (Engel, intrinsic form).** Over every field, \(\mathfrak g\) is nilpotent if and only if each \(\operatorname{ad}x\) is nilpotent. **Proof.** If \(C^{c+1}\mathfrak g=0\), then \((\operatorname{ad}x)^c\mathfrak g\subseteq C^{c+1}\mathfrak g=0\). Conversely, apply Theorem 2.1 to \(\operatorname{ad}\mathfrak g\) acting on \(\mathfrak g\). In a strictly triangular basis every product of \(d=\dim\mathfrak g\) adjoint operators is zero. Since \(C^{d+1}\mathfrak g\) is spanned by their values on vectors, it is zero. The zero algebra needs no argument. \(\square\) **Corollary 2.3.** Every nonzero ideal of a nilpotent algebra meets its centre nontrivially. In particular a nonzero nilpotent algebra has nonzero centre. **Proof.** For a nonzero ideal \(I\), set \(I_0=I\) and \(I_{j+1}=[\mathfrak g,I_j]\). These spaces eventually vanish, because \(I_j\subseteq C^{j+1}\mathfrak g\). The last nonzero one is killed by bracketing with \(\mathfrak g\), hence lies in \(I\cap Z(\mathfrak g)\). \(\square\) The representation matters. A one-dimensional abelian algebra is nilpotent, but it can act by the identity on \(k\), which is not a nilpotent operator. Engel's intrinsic criterion concerns its adjoint action. ## 3. Lie's theorem: a common eigenvector Strict triangularity required nilpotent operators. For solvable algebras the best possible conclusion is ordinary triangularity: diagonal characters may remain. **Theorem 3.1 (Lie).** Suppose \(k\) is algebraically closed of characteristic zero. If a solvable Lie algebra \(\mathfrak g\) acts on \(V\ne0\), it has a common eigenvector: \[ xv=\lambda(x)v\quad(x\in\mathfrak g) \] for some nonzero \(v\) and linear functional \(\lambda:\mathfrak g\to k\). The action is upper-triangular in a suitable basis. **Proof.** Induct on \(\dim\mathfrak g\), beginning with the zero algebra. For nonzero solvable \(\mathfrak g\), the ideal \([\mathfrak g,\mathfrak g]\) is proper; otherwise its derived series would never reach zero. Choose a codimension-one subspace \(\mathfrak a\) containing it. This is an ideal and is solvable. Induction yields \(v\ne0\) and \(\lambda:\mathfrak a\to k\) such that \(yv=\lambda(y)v\). Choose \(x\notin\mathfrak a\). Let \(W\) be the cyclic space spanned by \(v,xv,x^2v,\ldots\), and choose its basis \(v,xv,\ldots,x^{r-1}v\), stopping at the first dependence. This basis exists because \(V\) is finite-dimensional. The space \(W\) is \(x\)-invariant. For \(y\in\mathfrak a\), induction on \(j\) gives \[ yx^jv-\lambda(y)x^jv \in\operatorname{span}\{v,xv,\ldots,x^{j-1}v\}. \tag{3.1} \] The case \(j=0\) is the choice of \(v\). For the step, write \(yx^jv=x(yx^{j-1}v)+[y,x]x^{j-1}v\). The inductive assertion applies also to \([y,x]\in\mathfrak a\), so the second term lies in the indicated lower span; the first has leading term \(\lambda(y)x^jv\). Thus \(W\) is \(\mathfrak a\)-invariant and every \(y|_W\) has all diagonal entries \(\lambda(y)\). Both \(x\) and \(y\) preserve \(W\), so the trace of their commutator is zero. Applying (3.1) to \([x,y]\) gives \[ 0=\operatorname{tr}([x,y]|_W)=r\lambda([x,y]). \] Characteristic zero implies \(\lambda([x,y])=0\). Now consider the entire simultaneous eigenspace \[ E_\lambda=\{w:yw=\lambda(y)w\text{ for all }y\in\mathfrak a\}. \] It is nonzero and \(x\)-invariant, because for \(w\in E_\lambda\), \(y(xw)=\lambda(y)xw+[y,x]w=\lambda(y)xw\). An operator on a nonzero finite-dimensional space over an algebraically closed field has an eigenvector. Choose one for \(x|_{E_\lambda}\); it is a common eigenvector for \(\mathfrak a+kx=\mathfrak g\). Its line is invariant. Repeat the theorem on the quotient and lift the resulting flag to \(V\). A flag-adapted basis makes all operators upper-triangular. \(\square\) Any common eigenvalue functional vanishes on \([\mathfrak g,\mathfrak g]\), since scalars commute. An irreducible nonzero finite-dimensional representation of a solvable algebra under these hypotheses is therefore one-dimensional: its common eigenvector spans a nonzero invariant subspace. **Corollary 3.2.** Over every field of characteristic zero, \(\mathfrak g\) is solvable if and only if \([\mathfrak g,\mathfrak g]\) is nilpotent. **Proof.** First suppose the field is algebraically closed. If \(\mathfrak g\) is solvable, triangularize its adjoint representation. Commutators of upper-triangular matrices are strictly upper-triangular. Thus for every \(z\in[\mathfrak g,\mathfrak g]\), its action on that derived ideal is nilpotent. Engel's intrinsic criterion makes the ideal nilpotent. For an arbitrary characteristic-zero field, extend to an algebraic closure and apply Proposition 1.2 to the algebra and its derived ideal. Conversely a nilpotent derived ideal is solvable, and the quotient by it is abelian; Proposition 1.1 makes \(\mathfrak g\) solvable over any field. \(\square\) ## 4. The radical collects all solvable ideals **Proposition 4.1.** There is a largest solvable ideal, denoted \(\operatorname{rad}(\mathfrak g)\). Its quotient has zero radical. **Proof.** If \(I,J\) are solvable ideals, \((I+J)/I\) is isomorphic to \(J/(I\cap J)\), so it is solvable. Proposition 1.1 shows that \(I+J\) is solvable. Induction gives the same conclusion for any finite sum. The sum \(R\) of all solvable ideals is already a finite sum: choose a basis of \(R\), write each basis vector using finitely many such ideals, and collect the resulting finite list. Thus \(R\) is a solvable ideal containing every other one. This proves existence and maximality. If \(\bar J\) is a solvable ideal in \(\mathfrak g/R\), its inverse image \(J\) is an ideal containing \(R\). The extension with solvable ideal \(R\) and quotient \(J/R=\bar J\) is solvable, so \(J\subseteq R\). Hence \(\bar J=0\). \(\square\) For the two-dimensional algebra above, the radical is the whole algebra even though its centre is zero. A radical is not a centre. The radical of \(\mathfrak b_n\) is likewise \(\mathfrak b_n\). ## 5. What the field conditions protect Over \(\mathbb R\), the abelian algebra spanned by \[ J=\begin{pmatrix}0&-1\\1&0\end{pmatrix} \] is solvable, but its action on \(\mathbb R^2\) has no real eigenvector: \(J^2=-1\), so a real eigenvalue would have square \(-1\). This is the rotation algebra \(\mathfrak{so}(I_2)\). Passing to \(\mathbb C\) supplies the missing eigenlines. In characteristic \(p>0\), take \(A=k[t]/(t^p)\). Differentiation \(D\) is well-defined because \(D(t^p)=0\); let \(T\) be multiplication by \(t\). Then \[ [D,T]=1,\qquad [D,1]=[T,1]=0. \] The span of \(D,T,1\) is a three-dimensional Heisenberg algebra, hence nilpotent and solvable. Its action on \(A\) is irreducible. From any nonzero polynomial of degree \(r