In the above, a boost can be represented as a 4 × 4 matrix. The boost matrix B(v) means the boost B that uses the components of v, i.e. v 1, v 2, v 3 in the entries of the matrix, or rather the components of v / c in the representation that is used in the section Lorentz transformation#Matrix forms. The matrix entries depend on the components of the 3-velocity v, and that's what the notation B(v) means. It could be argued that the entries depend on the components of the 4-velocity because 3 of the entries of the 4-velocity are the same as the entries of the 3-velocity, but the usefulness of parameterizing the boost by 3-velocity is that the resultant boost you get from the composition of two boosts uses the components of the 3-velocity composition u ⊕ ⊕ v in the 4 × 4 matrix B(u ⊕ ⊕ v). But the resultant boost also needs to be multiplied by a rotation matrix because boost composition (i.e. the multiplication of two 4 × 4 matrices) results not in a pure boost but a boost and a rotation, i.e. a 4 × 4 matrix that corresponds to the rotation Gyr[ u, v ] to get B(u)B(v) = B(u ⊕ ⊕ v)Gyr[ u, v ] = Gyr[ u, v ]B(v ⊕ ⊕ u).