The Nitty Gritty Details

Note that

n^T l	=	`al`_x + `bl`_y + `cl`_z + `dl`_w
n^T s	=	`as`_x + `bs`_y + `cs`_z + `d`

so we can write p' = (n^Tl)s − (n^Ts)l in component form as

`x`'	=	(`al`_x + `bl`_y + `cl`_z + `d`) `s`_x	−	(`as`_x + `bs`_y + `cs`_z + `d`) `l`_x
`y`'	=	(`al`_x + `bl`_y + `cl`_z + `d`) `s`_y	−	(`as`_x + `bs`_y + `cs`_z + `d`) `l`_y
`z`'	=	(`al`_x + `bl`_y + `cl`_z + `d`) `s`_z	−	(`as`_x + `bs`_y + `cs`_z + `d`) `l`_z
`w`'	=	(`al`_x + `bl`_y + `cl`_z + `d`)	−	(`as`_x + `bs`_y + `cs`_z + `d`) `l`_w