Not sure about "elementary" but, at least, he tried to reduce it to "the simplest and most intelligible form". He assumes the level of a graduating secondary school student, and warns of the "patience and force of will" required to get through.
The linked article refers to the German 1916 publication of Über die spezielle und allgemeine Relativitätstheorie, 1920 reprint is available on the Internet Archive. The first English translation, by Lawson, Relativity: the Special and the General Theory, first appeared also in 1920, and is available on ibiblio.org. Here is Einstein's preface in full:
"The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of
clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a “step-motherly” fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for trees. May the book bring someone a few happy hours of suggestive thought!"