This paper presents an improvement on Cluster Treatment of Characteristic Roots (CTCR), which is a well-known methodology for delay-dependent stability analysis of multiple time delay systems (MTDS). We propose an algebraic approach to extract the stability switching hypersurfaces in spectral delay space, instead of a numerical procedure in CTCR with Extended Kronecker Sum (EKS) operation. The proposed algebraic approach is based on an efficient zero location test, and the deployment of this test to an auxiliary characteristic polynomial whose unique properties have recently been revealed. The achieved improvement is demonstrated by applying the new CTCR procedure to a system with three delays. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.