An extremal element in a Lie algebra $\mathfrak{g}$ over a field of characteristic not 2 is an element $x\in \mathfrak{g}$ such that $[x,[x,\mathfrak{g}]]$ is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of $\mathfrak{g}$, i.e. a subspace I satisfying $[I,[I,\mathfrak{g}]]\le I$. We show that in characteristic different from $2,3$ the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set.

This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from $2,3$ each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra $\mathfrak{g}$ and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.

李代数中的极值元素 $\mathfrak{G}$ 在特征不是2的字段上是一个元素 $X\in \mathfrak{G}$ 这样 $[X，[X，\mathfrak{G}]]$包含在x的线性范围内。极值元素的线性跨度（称为极值点）是$\mathfrak{G}$，即我满足的子空间$[\mathrm{一世}，[\mathrm{一世}，\mathfrak{G}]]\le \mathrm{一世}$。我们显示出与$\mathrm{2个}，3$ 具有点的几何设置了一组极点，而包含至少两个极点的最小内部理想作为线是Moufang球形建筑物，或者在没有线的情况下设置了Moufang集。

在Moufang集上的最后结果是通过将Lie代数连接到可构造代数而获得的，可构造代数是一类具有对合广义Jordan代数的非缔合代数。结果表明，在特性上与$\mathrm{2个}，3$由极值元素生成的每个有限维简单Lie代数要么是辛Lie代数，要么可以通过将Tits-Kantor-Koecher构造应用于偏维一可构造代数来获得。李代数之间的各种关系$\mathfrak{G}$ 一方面研究了其极值几何，另一方面研究了相关的可构造代数。