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Significant Accounting Policies - Goodwill and Other Intangibles (Details)
 3 Months Ended 12 Months Ended
May 31, 2025
USD ($)
Aug. 31, 2024
USD ($)
intangible_asset
Aug. 31, 2025
USD ($)
Aug. 31, 2024
USD ($)
Aug. 31, 2023
USD ($)
intangible_asset
Jun. 01, 2025
USD ($)
intangible_asset
Jun. 01, 2024
USD ($)
intangible_asset
Jun. 01, 2023
Goodwill [Roll Forward]                
Beginning balance     $ 1,098,700,000 $ 1,097,900,000        
Additions from acquired businesses     394,600,000          
Foreign currency translation adjustments     2,200,000 800,000        
Ending balance   $ 1,098,700,000 1,495,500,000 1,098,700,000 $ 1,097,900,000      
Gross Carrying Amount   693,400,000 1,426,500,000 693,400,000        
Accumulated Amortization   (385,500,000) (460,100,000) (385,500,000)        
Indefinite-lived trade names           $ 132,500,000 $ 135,500,000  
Amortization expense of finite-lived intangible assets     76,500,000 39,700,000 42,100,000      
Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]                
2026     91,700,000          
2027     90,200,000          
2028     85,700,000          
2029     84,800,000          
2030     81,900,000          
Goodwill impairment charge         $ 0      
Indefinite-lived intangible assets held | intangible_asset           8 8  
Indefinite-lived intangible assets           $ 132,500,000 $ 135,500,000  
Intangible asset impairment charge $ 14,700,000              
Trade Names                
Goodwill [Roll Forward]                
Indefinite-lived trade names   132,600,000 132,600,000 132,600,000        
Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]                
Indefinite-lived intangible assets   $ 132,600,000 132,600,000 132,600,000        
Number of impaired intangible assets | intangible_asset   1     6      
Intangible asset impairment charge   $ 3,000,000.0     $ 14,000,000.0      
No longer indefinite lives | intangible_asset         5      
Number of unimpaired intangible assets | intangible_asset         7      
Developed technology and patents                
Goodwill [Roll Forward]                
Gross Carrying Amount   157,500,000 616,700,000 157,500,000        
Accumulated Amortization   (133,300,000) (170,500,000) (133,300,000)        
Trademarks and trade names                
Goodwill [Roll Forward]                
Gross Carrying Amount   45,500,000 176,900,000 45,500,000        
Accumulated Amortization   (20,500,000) (29,000,000.0) (20,500,000)        
Distribution networks                
Goodwill [Roll Forward]                
Gross Carrying Amount   61,800,000 61,800,000 61,800,000        
Accumulated Amortization   (51,600,000) (53,700,000) (51,600,000)        
Customer relationships                
Goodwill [Roll Forward]                
Gross Carrying Amount   428,600,000 571,100,000 428,600,000        
Accumulated Amortization   (180,100,000) (206,900,000) (180,100,000)        
Trade Names No Longer With Indefinite Lives                
Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]                
Amortization period               15 years
ABL                
Goodwill [Roll Forward]                
Beginning balance     1,015,100,000 1,014,400,000        
Additions from acquired businesses     0          
Foreign currency translation adjustments     900,000 700,000        
Ending balance   1,015,100,000 1,016,000,000 1,015,100,000 $ 1,014,400,000      
AIS                
Goodwill [Roll Forward]                
Beginning balance     83,600,000 83,500,000        
Additions from acquired businesses     394,600,000          
Foreign currency translation adjustments     1,300,000 100,000        
Ending balance   $ 83,600,000 $ 479,500,000 $ 83,600,000 $ 83,500,000