A symplectic polarity of a building {\Delta} of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type F4). In this paper, we show in a geometric way that every building of type E6 contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type F4 fully embedded in the natural point-line geometry of {\Delta} arises from a symplectic polarity.

中文翻译：

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将 $$\mathsf {F_4}$$F4 类型的建筑物拆分为 $$\mathsf {E_6}$$E6 类型的建筑物

E6 型建筑物{\Delta} 的辛极性是其不动点结构为F4 型建筑物的极性，该建筑物包含与辛极性空间同构的残基（即所谓的F4 型分裂建筑物）。在本文中，我们以几何方式展示了每个 E6 类型的建筑物都包含，直到共轭，一类独特的辛极性。我们还表明，完全嵌入 {\Delta} 的自然点线几何中的每个 F4 型分裂建筑物的自然点线几何来自辛极性。