A special conformal transformation is a conformal transformation whose finite Euclidean form in one sign convention is wherever the denominator is nonzero. Its position-dependent scale change preserves local angles.
In three boundary dimensions, the three parameters supplement translations, rotations and dilation to form the ten-dimensional de Sitter isometry algebra. The full transformation of the flat-slicing spacetime also acts on conformal time; the boundary formula alone is not that bulk transformation. A matter equation with a propagation speed different from the metric speed of light generally need not preserve these isometries.
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Special conformal transformations are a specific type of transformation that can be applied in the context of conformal field theories (CFTs) and conformal geometry. In a conformal transformation, angles are preserved, but distances may change. Special conformal transformations are a special subset of these transformations that involve a specific modification of the space-time coordinates.