underBruhat(BasicList) -- Weyl group elements just under the ones in the list for the Bruhat order

i1 : R=rootSystemA(3)

o1 = RootSystem{...8...}

o1 : RootSystem

i2 : L=underBruhat(longestWeylGroupElement(R))

o2 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, |  2 |},
                                             | -2 |   | -1 |  
                                             |  1 |   |  0 |  
     ------------------------------------------------------------------------
     {WeylGroupElement{RootSystem{...8...}, | -2 |}, | -1 |},
                                            |  1 |   |  2 |  
                                            | -2 |   | -1 |  
     ------------------------------------------------------------------------
     {WeylGroupElement{RootSystem{...8...}, |  1 |}, |  0 |}}
                                            | -2 |   | -1 |
                                            | -1 |   |  2 |

o2 : List

i3 : L1=apply(L,x->x#0)

o3 = {WeylGroupElement{RootSystem{...8...}, | -1 |},
                                            | -2 |  
                                            |  1 |  
     ------------------------------------------------------------------------
     WeylGroupElement{RootSystem{...8...}, | -2 |},
                                           |  1 |  
                                           | -2 |  
     ------------------------------------------------------------------------
     WeylGroupElement{RootSystem{...8...}, |  1 |}}
                                           | -2 |
                                           | -1 |

o3 : List

i4 : underBruhat(L1)

o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
                                             |  2 |        | -1 |       |  1
                                             | -3 |        |  2 |       |  1
     ------------------------------------------------------------------------
     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
     |                                            | -3 |        | -1 |      
     |                                            |  1 |        |  2 |      
     ------------------------------------------------------------------------
     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
     | -1 |                                            | -1 |        |  1 |  
     |  0 |                                            | -2 |        |  1 |  
     ------------------------------------------------------------------------
     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
         |  2 |                                            |  2 |        |  1
         | -1 |                                            | -1 |        | -1
     ------------------------------------------------------------------------
     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
     |       | -1 |                                            | -1 |       
     |       |  0 |                                            |  2 |       
     ------------------------------------------------------------------------
     | -1 |}, {1, |  1 |}}}}
     |  2 |       |  1 |
     | -1 |       | -1 |

o4 : List