Summary: An inner ideal of a Lie algebra \(L\) over a commutative ring \(k\) is a \(k\)-submodule \(B\) of \(L\) such that \([B[BL]] \subseteq B\). This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let \(L\) be a simple Lie algebra in which \(D_y^2 = 0\) implies \(y = 0\), where \(D_y\) denotes the adjoint mapping determined by \(y\). If \(L\) satisfies the descending chain condition on inner ideals and has proper inner ideals, then \(L\) contains a subalgebra \(S = \langle e,f,h\rangle \), isomorphic to the split 3-dimensional simple Lie algebra, such that \(D_e^3 = D_f^3 = 0\). Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when \(L\) is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic \(p > 5\) this decomposition yields:
Theorem. \(L\) is classical if and only if there is an \(x \neq 0\) in \(L\) such that \(D_x^{p-1} = 0\) and if \(D_y^2 = 0\) implies \(y = 0\). The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.