I found this description in the book on Kac Moody Algebra by Wan. Can someone explain whats meant by triple?

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Title: Realization of a Complex nXn Matrix A

We start with an arbitrary complex n X n matrix of rank ; and define a realization of A to be the triple where is a complex vector space of finite dimension, and are indexed subsets of and respectively such that they satisfy following properties

1) Both sets are linearly independent

2)

where denotes the pairing

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