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Significant Accounting Policies - Goodwill and Other Intangibles (Details)
 3 Months Ended 12 Months Ended
Jun. 01, 2024
USD ($)
intangible_asset
Aug. 31, 2024
USD ($)
intangible_asset
reporting_unit
Aug. 31, 2024
USD ($)
Aug. 31, 2023
USD ($)
reporting_unit
intangible_asset
Aug. 31, 2022
USD ($)
reporting_unit
Jun. 01, 2023
Goodwill [Roll Forward]            
Beginning balance     $ 1,097,900,000 $ 1,084,300,000    
Additions from acquired businesses       15,200,000    
Adjustments to provisional amounts from acquired businesses       (200,000)    
Derecognitions for divestitures       (700,000)    
Foreign currency translation adjustments     800,000 (700,000)    
Ending balance $ 1,100,000,000 $ 1,098,700,000 1,098,700,000 1,097,900,000 $ 1,084,300,000  
Gross Carrying Amount   693,400,000 693,400,000 691,100,000    
Accumulated Amortization   (385,500,000) (385,500,000) (345,500,000)    
Indefinite-lived trade names $ 135,500,000          
Amortization expense of finite-lived intangible assets     39,700,000 42,100,000 41,000,000.0  
Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]            
2025   32,100,000 32,100,000      
2026   29,800,000 29,800,000      
2027   28,300,000 28,300,000      
2028   24,200,000 24,200,000      
2029   $ 22,800,000 22,800,000      
Goodwill impairment charge       $ 0 $ 0  
Number of reporting units | reporting_unit   3   3 3  
Indefinite-lived intangible assets held | intangible_asset 8          
Indefinite-lived intangible assets $ 135,500,000          
Number of unimpaired intangible assets | intangible_asset 7          
Impairment, Intangible Asset, Statement of Income or Comprehensive Income [Extensible Enumeration]       Special charges    
Trade Names            
Goodwill [Roll Forward]            
Indefinite-lived trade names   $ 132,600,000 132,600,000 $ 135,600,000    
Finite-Lived Intangible Assets, Net, Amortization Expense, Fiscal Year Maturity [Abstract]            
Indefinite-lived intangible assets   $ 132,600,000 132,600,000 $ 135,600,000    
Number of unimpaired intangible assets | intangible_asset       7    
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    
Patents and patented technology            
Goodwill [Roll Forward]            
Gross Carrying Amount   157,500,000 157,500,000 $ 158,800,000    
Accumulated Amortization   (133,300,000) (133,300,000) (122,300,000)    
Trademarks and trade names            
Goodwill [Roll Forward]            
Gross Carrying Amount   45,500,000 45,500,000 45,500,000    
Accumulated Amortization   (20,500,000) (20,500,000) (18,400,000)    
Distribution networks            
Goodwill [Roll Forward]            
Gross Carrying Amount   61,800,000 61,800,000 61,800,000    
Accumulated Amortization   (51,600,000) (51,600,000) (49,400,000)    
Customer relationships            
Goodwill [Roll Forward]            
Gross Carrying Amount   428,600,000 428,600,000 425,000,000.0    
Accumulated Amortization   (180,100,000) (180,100,000) (155,400,000)    
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     1,014,400,000 1,014,200,000    
Additions from acquired businesses       0    
Adjustments to provisional amounts from acquired businesses       0    
Derecognitions for divestitures       (700,000)    
Foreign currency translation adjustments     700,000 900,000    
Ending balance   1,015,100,000 1,015,100,000 1,014,400,000 $ 1,014,200,000  
ISG            
Goodwill [Roll Forward]            
Beginning balance     83,500,000 70,100,000    
Additions from acquired businesses       15,200,000    
Adjustments to provisional amounts from acquired businesses       (200,000)    
Derecognitions for divestitures       0    
Foreign currency translation adjustments     100,000 (1,600,000)    
Ending balance   $ 83,600,000 $ 83,600,000 $ 83,500,000 $ 70,100,000