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Nonsolvable Subalgebras of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="fraktur">g</mml:mi><mml:mi mathvariant="fraktur">l</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">4</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>
All the simple and then semisimple subalgebras of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi mathvariant="fraktur">g</mml:mi><mml:mi mathvariant="fraktur">l</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">4</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> are found. Each such semisimple subalgebra acts by commutator on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi mathvariant="fraktur">g</mml:mi><mml:mi mathvariant="fraktur">l</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">4</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. In each case the invariant subspaces are found and the results are used to …