æz ðə kˌɔːɹɪspˈɑːndɪŋ pˈɑːzᵻtˌɪv nˈɪlpoʊtənt sˈʌbɐldʒˌɛbɹə ænd nˈɛɡətˌɪv nˈɪlpoʊtənt sˈʌbɐldʒˌɛbɹɚ ɹᵻspˈɛktɪvli, dˈuː tə ðə pˌɔɪŋkɑːɹɹˈeɪ bˈɜːkhɔf wˈɪt θˈiəɹəm ðɛɹ ɪz ɐ dˌiːkɑːmpəzˈɪʃən
as the corresponding positive nilpotent subalgebra and negative nilpotent subalgebra respectively, due to the Poincaré–Birkhoff–Witt theorem there is a decomposition