Each 2-diagonal (joining vertices two steps distant along the perimeter of the octagon) must intersect with 2 others of its kind, plus two 3-diagonals and one 4-diagonal, all emanating from the vertex between on the perimeter.
There are 8 such 2-diagonals, so multiply the previous numbers by 8, except for the intersections of 2-diagonals with their own kind: that number only by 4.
Each 3-diagonal must intersect with two 2-diagonals (but these are already counted), plus four 3-diagonals and both 4-diagonals, all emanating from the two vertices between on the perimeter.
There are 8 such 3-diagonals, so multiply accordingly.
That leaves intersections with 4-diagonals that are not already counted, which is those with their own kind. Each must intersect all 3 others.