The rows are out of order
In[1]:= kl = {
{1, 0, 0, 0, 0, 0},
{-1, -6, -2, 0, 0, 0},
{0, -2, -8, -2, 0, 0},
{0, 0, -2, 8, 6, 0},
{0, 0, 0, 6, -4, -8},
{0, 0, 0, 0, 0, 1}};
{lu, p, c} = LUDecomposition[kl];
l = lu SparseArray[{i_, j_} /; j < i -> 1, {6, 6}] + IdentityMatrix[6];
u = lu SparseArray[{i_, j_} /; j >= i -> 1, {6, 6}];
l.u
Out[5]= {
{1, 0, 0, 0, 0, 0},
{0, -2, -8, -2, 0, 0},
{0, 0, -2, 8, 6, 0},
{0, 0, 0, 6, -4, -8},
{-1, -6, -2, 0, 0, 0},
{0, 0, 0, 0, 0, 1}}