Hyperspectral unmixing (HU) is an essential signal processing procedure for blindly extracting the hidden spectral signatures of materials (or endmembers) from observed hyperspectral imaging data. Craig's criterion, stating that the vertices of the minimum volume enclosing simplex (MVES) of the data cloud yield high-fidelity endmember estimates, has been widely used for designing endmember extraction algorithms (EEAs) especially in the scenario of no pure pixels. However, most Craig-criterion-based EEAs generally suffer from high computational complexity due to heavy simplex volume computations, and performance sensitivity to random initialization, etc. In this work, based on the idea that Craig's simplex with N vertices can be defined by N associated hyperplanes, we develop a fast and reproducible EEA by identifying these hyperplanes from N(N - 1) data pixels extracted via simple and effective linear algebraic formulations, together with endmember identifiability analysis. Some Monte Carlo simulations are provided to demonstrate the superior efficacy of the proposed EEA over state-of-the-art Craig-criterion-based EEAs in both computational efficiency and estimation accuracy.