The action of a 50:50 beam splitter in quantum optics can be represented by a unitary matrix that describes the transformation of input modes to output modes. The two matrices you mentioned correspond to different conventions for representing the beam splitter transformation.
The matrix B = (1/sqrt[2]) (1 I I 1) is commonly referred to as the "symmetric" representation, while the matrix B = (1/sqrt[2]) (1 1 1 -1) is known as the "anti-symmetric" representation.
Both representations are valid and equivalent in terms of describing the action of a 50:50 beam splitter. The only difference is how the input and output modes are labeled and ordered. In the symmetric representation, the input modes are labeled I and the output modes are labeled 1 and 2. In the anti-symmetric representation, the input modes are labeled 1 and 2, and the output modes are also labeled 1 and 2.
The choice of representation may vary depending on the context or convention used in different sources, such as research papers or textbooks. It's important to be aware of these different conventions and understand the corresponding matrix representations when studying or working with quantum optics experiments involving beam splitters.