I'm reading "Symmetric Bilinear Forms" by Milnor. In it he makes the following definition.
"A symmetric inner product space $S$ over the ring $R$ is said to be split if there exists a submodule $N\subset S$ such that $N$ is a direct summand of $S$, and such that $N$ is precisely equal to its orthogonal complement $N^{\bot}$."
Is this correct, "precisely equal to its orthogonal complement"? seems off. Maybe I just don't understand it. He also mentions "this concept is due to Knebusch, who uses the term "metabolic" in place of our split".
Would be great if anyone could clear this up or point me to another reference.