Let g be a semi-simple Lie algebra. A Borel subalgebra  b is any maximal solvable subalgebra. A parabolic subalgebra p is any subalgebra containing a Borel subalgebra b. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form 

Let h be an Cartan subalgebra and  the associated root space decomposition. Let be a choice of positive roots and let be a set of simple roots. The subalgebra is called the standard Borel subalgebra associated to h and any parabolic subalgebra containing it is called a standard parabolic subalgebra.  (One could replace the summation over the positive roots by one over the negative roots.)

 Given a standard parabolic subalgebra p , let  This set of simple roots completely specifies the parabolic subalgebra p. Conversely, given a set of simple roots   let   is a linear combination of the roots in and set . Then is a standard parabolic subalgebra.

For the parabolic subalgebras of a real semi-simple Lie algebra the situation is essentially the same except that one must consider the restricted root space decomposition  relative to a maximal Abelian subalgebra a on which the Killing form is positive-definite.

Let Σ be a subset of the simple roots  and set   =    The command ParabolicSubalgebra returns the standard parabolic subalgebra  The command ParabolicSubalgebraRoots returns the list of simple roots

With the keyword argument method  = "non-compact",  a real parabolic subalgebra is calculated.

With  the standard Borel subalgebra is returned.

If the Lie algebra is created from the command SimpleLieAlgebraData , then the table obtained from the command SimpleLieAlgebraProperties can be used as the second argument or  

The command Query/"ParabolicSubalgebra" will test if a given subalgebra of a semi-simple Lie algebra is parabolic.