Let $A$ be a left symmetric algebra over a real Lie algebra $\grg$. A symmetric bilinear form $\dyk{\enskip,\enskip}$ of $A$ is called {\it of Hessian type} if the following equality holds: \[\dyk{xy,z}+\dyk{y,xz}=\dyk{yx,z}+\dyk{x,yz}\quad(x,y,z\in A).\] H. Shima studied the structures of left symmetric algebras with a positive definite symmetric bilinear form of Hessian type ([S]). Denote by $h$ the bilinear form defined by \[h(x,y)=\Tr R(xy),\] where $R(x)$ denotes the right multiplication on $A$ by an element $x$. $h$ is a symmetric bilinear form of Hessian type. \\ It is called {\it the canonical 2-form on $A$}. A left symmetric algebra $A=(\grg,e,h)$ over $\grg$ with a right identity $e$ and the non degenerate canonical 2-form $h$ is called {\it regular}. In this paper, we shall study the structures of a regular algebra over a real reductive Lie algebra and related topics.