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Acquisitions and Divestitures (Details)
$ in Millions
1 Months Ended 12 Months Ended
Nov. 30, 2022
USD ($)
Aug. 31, 2023
USD ($)
Aug. 31, 2022
USD ($)
acquisition
Oct. 30, 2022
USD ($)
Aug. 31, 2021
USD ($)
Business Acquisition [Line Items]          
Number of acquisitions | acquisition     0    
Cash outflows for working capital settlements     $ (12.9)    
Goodwill   $ 1,097.9 $ 1,084.3   $ 1,094.7
Disposal Group, Not Discontinued Operation, Gain (Loss) on Disposal, Statement of Income or Comprehensive Income [Extensible Enumeration]   Other Nonoperating Income (Expense)      
Intelligent Spaces Group Segment          
Business Acquisition [Line Items]          
Goodwill   $ 15.0      
Disposal Group, Not Discontinued Operations | Sunoptics Business          
Business Acquisition [Line Items]          
Assets transferred       $ 15.1  
Pre-tax loss on sale $ 11.2 11.2      
Fiscal 2023 Acquisitions          
Business Acquisition [Line Items]          
Goodwill   15.0      
Identifiable intangible assets   $ 18.0      
Weighted average amortization period in years   15 years      
Fiscal 2021 Acquisitions          
Business Acquisition [Line Items]          
Goodwill   $ 12.3      
Identifiable intangible assets   $ 6.7      
Weighted average amortization period in years   11 years      
Goodwill expected to be tax deductible   $ 9.2