In a previous paper [EG] we described an integral structure (J, E) on the exceptional Jordan algebra of Hermitian 3 x 3 matrices over the Cayley octonions. Here we use modular forms and Niemeier's classification of even unimodular lattices of rank 24 to further investigate J and the integral, even lattice J(0) = (ZE)(perpendicular to) in J. Specifically, we study ring embeddings of totally real cubic rings A into J which send the identity of A to E, and we give a new proof of R. Borcherds's result that J(0) is characterized as a Euclidean lattice by its rank, type, discriminant, and minimal norm. [References: 18]
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机译：在先前的论文[EG]中，我们描述了Cayley octonion上Hermitian 3 x 3矩阵的非凡Jordan代数上的积分结构（J，E）。在这里，我们使用模数形式和秩为24的偶数单模格的Niemeier分类进一步研究J和J中的积分偶数J（0）=（ZE）（垂直于）。具体而言，我们研究完全实立方的环嵌入环A到J，将A的身份发送给E，我们给出了R的新证明。Borcherds的结果是，J（0）通过其秩，类型，判别和极小范数被表征为欧几里得格。 [参考：18]
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