Skew transformation inverse Issue when constructed with axes Method

Developers Ruby API
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Did you consider that SketchUp stores in the last matrix element (m[15]) a uniform scale factor (scales in all directions)? To normalize the matrix I believe one has to do:

[ 0, 1, 2].each{ |i| m[i] /= m[15] }
[ 4, 5, 6].each{ |i| m[i] /= m[15] }
[ 8, 9,10].each{ |i| m[i] /= m[15] }
# [12,13,14] store the translation which is unaffected.
m[15] = 1.0

I observed another inconsistency. Does the problem already occur in the Transformation.axes factory method or only in inverse? I recreated the skew transformation with the normal constructor and a matrix array:

skew2 = Geom::Transformation.new([ 1,0,0,0,  Math.sqrt(0.5),Math.sqrt(0.5),0,0, 0,0,1,0, 0,0,0,1 ])

which prints just like skew

puts tmatrix_bycolumn(skew2)
      1.000,    0.707,    0.000,    0.000
      0.000,    0.707,    0.000,    0.000
      0.000,    0.000,    1.000,    0.000
      0.000,    0.000,    0.000,    1.000

but has a different inverse:

puts tmatrix_bycolumn(skew2.inverse)
      1.000,   -1.000,    0.000,   -0.000
     -0.000,    1.414,   -0.000,   -0.000
      0.000,   -0.000,    1.000,   -0.000
      0.000,    0.000,    0.000,    1.000

which leads to the correct identity:

puts tmatrix_bycolumn(skew2 * skew2.inverse)
      1.000,    0.000,    0.000,    0.000
      0.000,    1.000,    0.000,    0.000
      0.000,    0.000,    1.000,    0.000
      0.000,    0.000,    0.000,    1.000

So a transformation created from axes must have a certain private attribute since it behaves different from a transformation with identical elements.