SAE was originally implemented for use between peers in IEEE 802.11s. When peers discover each other they take part in an SAE exchange. If SAE completes successfully, each peer knows the other party possesses the mesh password and, as a by-product of the SAE exchange, the two peers establish a cryptographically strong key. This key is used with the "Authenticated Mesh Peering Exchange" to establish a secure peering and derive a session key to protect mesh traffic, including routing traffic.
In 2019 Eyal Ronen and Mathy Vanhoef released an analysis of WPA3's Dragonfly handshake and found that "an attacker within range of a victim can still recover the password" and the bugs found "allow an adversary to impersonate any user, and thereby access the Wi-Fi network, without knowing the user's password."