Option 1 states that both the Kailasanatha Temple at Ellora and the Shore Temple at Mamallapuram are examples of Nagara architecture. However, this is not the common element between the two temples. Nagara architecture is a style of temple architecture that originated in North India, whereas the Shore Temple represents the Dravidian style of temple architecture, which is predominant in South India. Therefore, option 1 is incorrect.

Option 2 states that both temples are carved out from solid rocks. This is indeed the common element between the two temples. Both the Kailasanatha Temple and the Shore Temple were sculpted out of monolithic rocks. This technique of carving temples out of solid rocks is a remarkable feat of ancient Indian architecture.

Option 3 suggests that both temples are from the Gupta period. However, this is incorrect. The Kailasanatha Temple at Ellora was built during the Rashtrakuta period, and the Shore Temple at Mamallapuram was constructed during the Pallava period. Therefore, option 3 is incorrect.

Option 4 proposes that both temples were built under the patronage of Pallava Kings. This is also incorrect as the Kailasanatha Temple at Ellora was built by the Rashtrakuta dynasty. Therefore,