The smallest split Cayley hexagon has two symplectic embeddings

It is well known that the smallest split Cayley generalized hexagon H(2) can be embedded into the symplectic space W(5,2), or equivalently, into the parabolic quadric Q(6,2). We establish a second way to embed H(2) into the same space and describe a computer proof of the fact that these are essentially the only two embeddings of this type.