Pfister's sixteen-square identity

In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Pfister around the same time. There are several versions, a concise one of which is

If all ${\displaystyle x_{i}}$ and ${\displaystyle y_{i}}$ with ${\displaystyle i>8}$ are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The ${\displaystyle u_{i}}$ are

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