I need to find the longest element in a list of elements of Weyl group. The way I do it is:

```
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner
```

Here g1 and g2 are two given subsets of the Weyl group I would like to make the above faster by using parallel computations. Is there some way in Sage or Python to do this? Thank you very much!

Edit: the full codes are:

```
def find_permutation2(L1, L2):
perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
assert [L2[i-1] for i in perm] == L1
r=perm.reduced_word()
return r
def LongestPermWInSBWInverseSA(W,A,B):
t1=StablizerOfTuple(A)
t2=StablizerOfTuple(sorted(B))
s=W.simple_reflections()
#print(t1,t2)
t3=[]
for i in t1:
t3.append(s[i])
t4=[]
for i in t2:
t4.append(s[i])
t5=find_permutation2(B,sorted(B))
w=W.one()
for i in t5:
w=w*s[i]
#print(t4,w,t3)
r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
r=r2
return r
def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2):
g1=W.subgroup(S1)
g2=W.subgroup(S2)
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner
return r
def SymmetricGroupActionOnListSi(i,L): # w=s_i, L=[a1,a2,...,an]
r1=[]
for j in L:
r1.append(j)
t1=r1[i-1]
r1[i-1]=r1[i]
r1[i]=t1
r=r1
return r
def StablizerOfTuple(A): # A is a list, weakly increasing
k = len(A)
r=[]
for i in [1..k-1]:
t1=SymmetricGroupActionOnListSi(i,A)
if t1==A:
r.append(i)
return r
A=[1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4]
B=[3, 3, 3, 5, 5, 5, 4, 4, 4, 6, 6, 6]
kk=len(A)
#print(kk)
typ='A'
W=SymmetricGroup(kk)
w=LongestPermWInSBWInverseSA(W,A,B)
w
```