Prove that `|2alpha+beta+gamma+deltaalphabeta+gammadeltaalpha+beta+gamma+delta2(alpha+beta)(gamma+delta)alphabeta(gamma+delta)+gammadelta(alpha+beta)alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0`

Explanation

consider the product
`|{:(1,,1,,0),(alpha+beta,,gamma+delta,,0),(alphabeta,,gammadelta,,0):}|xx |{:(1,,1,,0),(gamma+delta,,alpha+beta,,0),(gammadelta,,alphabeta,,0):}|`
`= |{:(2,,alpha+beta+gamma+delta,,gammadelta+alphabeta),(alpha+beta+gamma+delta,,2(alpha+beta)(gamma+delta),,alphabeta(gamma+delta)+gammadelta(alpha+beta)),(alphabeta+gammadelta,,alphabeta(gamma+delta)+gammadelta(alpha+beta),,2alphabetagammadelta):}|`
`=Delta`
hence `Delta=0` [being the product of two determinants each equal to0]